Enumeration of spanning trees in simplicial complexes

Transcription

1 U.U.D.M. Report 2009:13 Enumeration of spanning trees in simplicial complexes Anna Petersson Teknologie licentiatavhandling i matematik som framläggs för offentlig granskning den 15 juni, kl 13.15, Häggsalen, Ångströmlaboratoriet, Uppsala Department of Mathematics Uppsala University

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3 Enumeration of spanning trees in simplicial complexes Anna Petersson May 18,

4 Abstract The Kirchoff Matrix Tree Theorem states that the number of spanning trees in a graph G is equal to the absolute value of any cofactor of the Laplacian matrix of G. As the theory of simplicial complexes is a generalization of the theory of graphs one would suspect that there is a generalization of the notion of spanning trees to simplicial complexes, such that the number of spanning trees in a given simplicial complex is counted in a similiar way. We provide such a generalization and show that these trees, weighted with the number of elements of the torsion subgroup of their individual kth homological group, can be counted by the absolute value of the determinant of a certain submatrix of the Laplacian matrix of. This paper also gives an overview of the field of enumeration of trees using the Laplacian matrix. 2

5 Contents 1 Introduction Summary of the results Related results Contents of this paper Remarks on publication history Acknowledgements Basic definitions and notation Simplicial complexes Homology Relative homology Cohomology Matroids Special simplicial complexes Notation Spanning trees in simplicial complexes Background Definition of spanning trees based on homology Definition in the special case β k 1 ( ) = Spanning trees defined for any pure simplicial complex Definition of (k 1)-cobases Experimental methods Alternative notions of trees in simplicial complexes The trees of Harary and Palmer The trees of Masbaum and Vaintrob Enumeration theorem for spanning MV -trees The trees of Faridi Some lemmas valid for any simplicial complex k-bases and k-trees (k 1)-cobases and their properties Proof of the Matrix Tree Theorem 41 7 Special cases 49 8 Discussion 54 3

6 1 Introduction The main result of this paper is a generalization of Kirchoff s Matrix Tree Theorem to arbitrary simplicial complexes. In Kirchoff s Theorem the number of spanning trees in a given graph equals the absolute value of the determinant of any cofactor of the Laplacian matrix corresponding to the original graph. The objective of this paper is to find a definition of spanning trees in simplicial complexes, consistent with the usual spanning trees in graph theory, and such that the number of spanning trees in a simplicial complex is counted by the determinant of a submatrix of the Laplacian matrix corresponding to the complex. A secondary objective of this paper is to give an overview of various definitions of trees in hypergraphs. The definitions all generalize different properties of trees in graph theory, and such an overview may therefore contribute to further research via cross-fertilization. 1.1 Summary of the results The main subject is, however, a specific definition of spanning trees in simplicial complexes. From now on, let be a d-dimensional pure simplicial complex that is connected and let 0 < k d. A spanning k-dimensional tree of is a subset T k such that the homology group H k ( T ) {0} of T = T k { } and such that the Betti number of T is such that β k 1 ( T ) = β k 1 ( ). The definition of spanning k-dimensional trees turns out to be equivalent to the definition of k-bases in [24], where the k-bases are simply defined as a set of k-faces corresponding to a maximal set of linearly independent columns in the matrix [ ] of the boundary operator with respect to the standard bases. We show the following proposition. Proposition 1.1. Suppose is a d-dimensional pure and connected simplicial complex and let 0 < k d. Then T is a spanning k-dimensional tree of if and only if T is a k-base of. Recall that, for the Laplacian matrix corresponding to a connected graph on n vertices we obtain an invertible matrix by eliminating any row and its corresponding column, and hence the determinant of this submatrix is nonzero. However, the Laplacian matrices L = [ ][ ] corresponding to simplicial complexes of dimension two or higher will not have this property in general. We will therefore consider the dual notion of (k 1)-cobases, which can also be given both a definition via linear independence and a homological characterization. In [24], the definition of a (k 1)-cobase of is a set S of 4

7 (k 1)-faces of that corresponds to a maximal set of linear independent rows in the boundary operator matrix [ ]. The homological characterization is given in the following proposition. Proposition 1.2. Let be a d-dimensional pure and connected simplicial complex and let 0 < k d. Suppose S k 1. Then S is a (k 1)- cobase in if and only if H k (, S) H k ( ), (1.1) β k 1 (, S) = 0. (1.2) In the general case, where the original simplicial complex only has to be pure and connected, we have the Matrix Tree Theorem for Simplicial Complexes. Theorem 1.3 (Matrix Tree Theorem for Simplicial Complexes). Let be a d-dimensional pure and connected simplicial complex and let 0 < k d. Let T k ( ) be the family of spanning k-dimensional trees in and let S be a (k 1)-cobase of. Then where T k T k H k 1 ( Tk ) 2 = Q = [Hk 2( )] 2 Q 2 [H k 2 ( S)] det(l S,S) 2 Z k 1 ( ) Z k 1 ( ) B k 1 ( ; Q) + Z k 1 ( S ). The proof has three main ingredients. Firstly we have the the Cauchy Binet Theorem [12], which generalizes the multiplicativity of the determinant the fact that the determinant of a product of two square matrices is equal to the product of the two determinants to non-square matrices. Instead of a simple product we get a sum of products of all the possible combinations of all the largest possible square matrices. Formally we have that if A is an m n matrix, B is an n m matrix and n > m then det(ab) = det(a [m],i ) det(b I,[m] ). I =m Secondly, we use a result from Lattice theory that concerns the size of factorial integer groups [5]. If M is a map from Z m to Z m, [M] is the corresponding matrix representation and M is such that det(m) 0, then the number of elements of the group Z m/ M(Z m ) equals the absolute value of det([m]). With this proposition from Lattice theory we prove Proposition 6.2, which 5

8 asserts that the absolute value of a certain determinant equals the number of elements of a relative homology group whenever T is a spanning tree in, and is zero otherwise. I.e., suppose S is a (k 1)-cobase of. Then det([ ] S,T ) 0 if and only if T is a spanning tree of and, furthermore, if T is a spanning tree of then det([ ] S,T ) = ± H k 1 ( T, S). Thirdly we use Lemma 6.6, proved by Lyons [24], which implies that H k 1 ( T, S) = Q [H k 2( S)] [H k 1 ( T )]. H k 2 ( )] We will also discuss three special cases since they have some special features and their proofs are simpler. The first is when is such that β k 1 ( ) = 0 and H k 2 ( ) {0}, the second when is such that β k 1 ( ) = 0 and β k 2 ( ) {0} and the third when is such that β k 1 ( ) = 0. In the last case we have the following theorem. Theorem 1.4. Let 0 < k d and let be a d-dimensional pure and connected simplicial complex such that β k 1 ( ) = 0. Let T k ( ) be the family of spanning k-dimensional trees in and let S k 1 be any (k 1)-cobase of. Furthermore, let S = k 1 \ S. Then T T k H k 1 ( T ) 2 = 1.2 Related results [H k 2( )] 2 [H k 2 ( S)] 2 det(l S,S) Russel s papers [24]and [25] are mainly concerned with random structures related to determinantal probability measures, where random trees of simplicial complexes are just a special case. A determinantal probability measure ([25], [24]) is a probability measure whose elementary cylinder probabilities are given by determinants. A fairly general construction is the following. Given any matrix M of rank r, we obtain a probability measure on the sets of r columns by defining the probability µ(b) of a subset B of columns of M as µ(b) = det(m B M B) 2 / det(mm ), (1.3) where M b denotes the submatrix of M consisting of the set B of columns of M. This is a probability measure by virtue of the Cauchy Binet Theorem. It is shown in [25] that elementary cylinder probabilities Pr(A B), where B is assumed to be distributed according to µ in (1.3), are given by the 6

9 determinant det(q A,A ), where Q is the matrix for the orthogonal projection onto the row space of M. Both the Kirchoff s Matrix Tree Theorem and the Matrix Tree Theorem for Simplicial Complexes are strongly related to determinantal probability measures. In case of simplicial complexes the determinantal probability measure corresponding to M = [ ] is the one where each base T is given a probability µ(t) proportional to the square of the number of elements in the torsion subgroup of H k ( T ). One common question is why simplicial complexes are useful. A strong motivation comes from the understanding of polytopes and other structures in combinatorial optimization. Every simplicial complex can be represented by a graph with the faces of as vertices and where the edges correspond to incidence relations. However, the spanning trees presented in this paper, are not the simplicial complexes corresponding to spanning trees in the graph representation of the original simplicial complex. When Gelfand asked Kalai [21] why is it that in combinatorics there is so much emphasis on graphs compared to higher dimensional objects? Kalai answered, among other things, that he personally liked the higher dimensional objects but that many of the miracles of graph theory fail at higher dimensions. It is true that many of the miracles of graph theory fail at higher dimensions. The spanning trees in simplicial complexes have generalizations of several properties from spanning trees in graph theory, but certainly not of every property. For example the trees in graph theory are collapsible, but that is not the case for the spanning trees presented here. Instead, one must expect that various properties from graph theory are useful in the context of different generalizations. For instance, the MV - trees, HP-trees, F-trees, and the trees presented in this paper all generalize different aspects of trees in graph theory, each useful for different purposes. Apart from the intrinsic values of the theory of simplicial complexes it also presents a bird s eye view of some of those miracles of graph theory. 1.3 Contents of this paper Section 2 presents basic definitions of simplicial complexes, homology groups and matroids, as well as notations used in this paper. Section 3 provides the definitions of spanning trees, bases and cobases in simplicial complexes together with an overview of earlier results and definitions. Section 4 presents the notions of trees in simplicial complexes as defined by Harary and Palmer [14], Masbaum and Vaintrob [26], Faridi [8] and some 7

10 related results. The trees presented in this section are not consistent with the spanning trees of Section 3 and are not essential for understanding the results of this paper. Section 5 contains some basic lemmas and propositions for spanning trees in pure connected simplicial complexes. Section 6 gives a proof of the Matrix Tree Theorem for pure and connected simplicial complexes. Section 7 gives a proof of the Matrix Tree Theorem in the special case β k 1 ( ) = 0. Section 8 discusses possible future directions for research in this field. 1.4 Remarks on publication history In 2007 I solved the special case when is such that β k 1 ( ) = 0 and H k 2 ( ) = {0}, where d is the dimension of and 0 < k d, right before going on leave. When I returned I started working on the general case. In 2008 Duval, Klivans and Martin published an article where they proved the special case when H i ( ; Q) = {0} if i < k. After reading their article I realized that with a minor change my proof in the special case could be extended to such that β k 1 ( ) = 0 and β k 2 ( ) = 0. During autumn of 2008 it turned out that I was not alone in working on the general case; in his work on determinantal probability measures Lyons had solved the general case. I had already solved part of it, but Lyons revealed the final pieces of the general case. After reading his work it also became clear that the special case could be further improved to such that β k 1 ( ) = Acknowledgements I would like to express my gratitude for many interesting ideas, concrete advise and moral support, to my supervisors Lars Svensson, Royal Institute of Technology, Anders Johansson, University of Gävle and Svante Janson, University of Uppsala. Many thanks also to my colleagues Anders Pelander, Kajsa Bråting, Anders Öberg, Inga-Lena Assarsson and Pierre Bäcklund. I thank the graduate school FMB for financial support. 8

11 2 Basic definitions and notation 2.1 Simplicial complexes The study of simplicial complexes started as a study of polytopes in R n. In practice it turned out that keeping track of the geometry became inconvenient in many cases. For many purposes it is enough to use the so called abstract simplicial complexes, which keeps the essential properties of simplicial complexes by a definition based on set theory rather than geometry. In this paper we will only consider abstract simplicial complexes and, for simplicity, we will refer to them as simplicial complexes. Let V be a finite set. A hypergraph is any subset of the power set of V. A simplicial complex is a subset of the set of all possible subsets of V (and hence it is a hypergraph) such that is closed under the operation of choosing a subset. In other words, is a subset of P(V ) such that { } and, if f and e f, then e. An element of a simplicial complex is called a face of. If σ V, then P(σ) is called a simplex. If σ = k + 1 then we say either that σ is a k-dimensional simplex, a k-simplex or a simplex of dimension k. A simplicial complex is said to be of dimension k, or is called a k- dimensional simplicial complex, if k is the largest dimension of any simplex in. A facet of a simplicial complex is a maximal face of, i.e., for all facets s in we have that if s S then s = S. A simplicial complex is said to be pure if all its facets have the same number of elements and it is said to be connected if the graph ( 0, 1 ) is connected. The subcomplex (j) = {s : s j + 1} is called the j- skeleton of. Both the set V and the subset of singletons of are referred to as the vertices of. Note that the j-skeleton, of any r-dimensional simplicial complex, is also a simplicial complex. Let k = {s : s = k + 1} where 1 k d. Now let T k. We define a useful subcomplex of { T = T k if 1 k d, T = otherwise. Example 2.1. Let G be a graph G = (V, E). Remember that E can be regarded as a subset of P 2 (V ). So G regarded as { } P 1 (V ) P 2 (V ) is a one-dimensional simplicial complex. 9

12 2.2 Homology Let σ be a p-face. Choose an ordering on the vertices of σ and let two orderings be equivalent if they differ by an even permutation. If p > 0 there are two equivalence classes of orderings (if p = 0 there is only one class) and both are called an orientation of σ. Let be an r-dimensional simplicial complex. If 0 p r then let C p ( ) be the free abelian group, with coefficients from Z, generated by the oriented p-faces of. Let σ = σ if σ and σ are generated by different orientations of the same p-face. If p < 0 or if p > r, then let C p ( ) be the trivial group {0}. The group C p ( ) is called the group of oriented p- chains and an element of it is called a p-chain. The boundary operator is a homomorphism p : C p ( ) C p 1 ( ), such that if σ = [v 0 v 1 v 2...v p ] is an oriented p-face then p (σ) = p ([v 0 v 1 v 2...v p ]) = p ( 1) i [v 0 v 1 v 2... ˆv i...v p ] i=0 where ˆv i means that v i is deleted. If p < 0 or p > d then let p be the trivial homomorphism. If p = 0 then let 0 : C 0 ( ) C 1 ( ) be the surjective homomorphism defined by ([v]) = 1[ ] for all {v} in 0. By the definition of simplicial complexes given in this paper, { }, and hence the homology in this paper is the reduced homology in other papers. Reduced homology is only a minor modification of homology, designed to make every point have all its reduced homology groups equal to the trivial group. But with the condition { } for every simplicial complex, the homology groups will be automatically reduced and H 0 ({v}) {0}. Note that H 0 ( ) Z ω 1 where ω is the number of components of, which is the number of components of the graph ( 0, 1 ). When there is no risk for confusion, we omit the subscript p and write only for the boundary operator. Regard the groups C p ( ) and C p 1 ( ) as vector spaces over Q and let [ p ] be the matrix corresponding to the map p : C p ( ) C p 1 ( ), with respect to the standard bases in both these vector spaces. In general, to define a homology, you need a chain complex, i.e., a sequence of abelian groups C 0, C 1, C 2,... which are connected by homomorphisms d 0, d 1, d 2,... such that d i : C i C i 1 and d i d i 1 (x) = 0, for all i in N. Note that the groups of oriented p-chains together with the boundary operators is a chain complex. 10

13 The group of p-cycles of is defined as Z p ( ) = ker( p ). Furthermore the group of p-boundaries of is defined as B p ( ) = im( p+1 ) = p+1 (C p+1 ( )). Note that B p ( ) Z p ( ). Now we are ready to define the pth homology group of H p ( ) = Z p ( )/B p ( ). where H p ( ) {0} if p < 0 or p > d. The pth betti number of is defined as β p ( ) = rank(h p ( )) (where rank refers to the torsion-free rank). Figure 2.1: The simplicial complex in Example 2.2. v 3 e 3 σ e 2 e 1 v 1 v 2 Example 2.2. To clarify the definitions, let us compute the chain groups and the homology groups for a simple simplicial complex seen in Figure 2.1. Let e 1 = v 2 v 1, e 2 = v 3 v 2, e 3 = v 1 v 2 and σ = v 3 v 2 v 1. i =, i < 0, i > 2 C i ( ) = {0}, i < 0, i > 2 0 = {v 1,v 2,v 3 } C 0 ( ) = v 1,v 2,v 3 1 = {e 1,e 2,e 3 } C 1 ( ) = e 1,e 2,e 3 2 = {σ} C 2 ( ) = σ Z i ( ) = {0}, i < 0, i 2 B i ( ) = {0}, i < 0, i 2 Z 0 ( ) = C 0 ( ) = v 1,v 2,v 3 B 0 ( ) = v 1 v 3,v 2 v 3 Z 1 ( ) = (e 1 + e 2 + e 3 ) B 1 ( ) = (e 1 + e 2 + e 3 ) H i ( ) = {0}, i < 0, i 2 β i ( ) = 0, i < 0, i 2 H 0 ( ) Z 3 /Z 2 Z β 0 ( ) = 1 H 1 ( ) = Z 1 /B 1 {0} β 1 ( ) = 0 Example 2.3. An example of a slightly larger simplical complex, which by the way will show up in discussions later in this paper. is the simplicial 11

14 Figure 2.2: The simplicial complex in Example 2.3. v 6 e 7 e 8 σ 3 v 2 e 9 v 4 e 1 e 2 e 4 e 5 σ 1 σ 2 v 1 e 3 v 3 e 6 v 5 complex seen in Figure 2.2. i =, i < 0,i > 2 0 = {v 1,v 2,v 3,v 4,v 5,v 6 } 1 = {e 1,e 2,e 3,e 4,e 5,e 6,e 7,e 8,e 9 } 2 = {σ 1,σ 2,σ 3 } C i ( ) = {0}, i < 0,i > 2 C 0 ( ) = v 1,v 2,v 3,v 4,v 5,v 6 Z 6 C 1 ( ) = e 1,e 2,e 3,e 4,e 5,e 6,e 7,e 8,e 9 C 2 ( ) = σ 1,σ 2,σ 3 Z i ( ) = {0}, i < 0, i 2 Z 0 ( ) = C 0 ( ) Z 1 ( ) = See the note. B i ( ) = {0}, i < 0, i 2 B 0 ( ) = v 1 v 6,v 2 v 6,v 3 v 6,v 4 v 6,v 5 v 6 B 1 ( ) = e 1 + e 2 + e 3,e 4 + e 5 + e 6,e 7 + e 8 + e 9 H i ( ) = {0}, i < 0 or i 2 β i ( ) = 0, i < 0 or i 2 H 0 ( ) Z 6 /Z 5 Z β 0 ( ) = 1 12

15 Since Z 1 is already a long expression (therefore it is omitted here) we use a trick to compute H 1 ( ). Suppose z = α 1 e α 9 e 9 Z 1 ( ), then we can rewrite this as z = γ 2 e 2 + γ 3 e 3 + γ 4 e 4 + γ 6 e 6 + γ 8 e 8 + γ 9 e 9 + [α 1 δ(σ 1 ) + α 5 δ(σ 2 ) + α 7 δ(σ 3 )], for some γ 2, γ 3, γ 4, γ 6, γ 8, γ 9 Z, where the part within the brackets belong to B 1 ( ). Since z is a cycle, we must have δ(z) = 0 and hence γ 3 = γ 6 = γ 8 = 0, γ 2 = γ 4 = γ 9. Therefore we have H 1 ( ) Z. Example 2.4. We also compute the homology of a graph. Let G = (V, E) be a graph. Then β 0 (G) equals the number of connected components of G, call it c, and β 1 (G) equals the number of cycles in G, call it z. Of course β 0 (G) = c 1, β 1 (G) = z and all other betti numbers are zero. Both H 0 (G) and H 1 (G) are free groups so H 0 (G) Z c, H 0 (G) Z c 1 and H 1 (G) = H 1 (G) Z z. Every other homology group is isomorphic to {0}. The homology we have defined so far is over the integers. In this paper, if not explicitly mentioned otherwise, the homology is always over Z. It is possible to compute the homology over other rings. If we let Q play the role of Z in the definition of homology we get the homology over Q and we will denote it by H p ( ; Q). It is defined analogously, H p ( ; Q) = Z p ( ; Q)/B p ( ; Q) where Z p ( ; Q) = ker p : C p ( ; Q) C p 1 ( ; Q) and B p ( ; Q) = im p+1 : C p+1 ( ; Q) C p ( ; Q). C i ( ; Q) are the vector spaces generated by the oriented p-simplices of with coefficients from Q instead of Z. In general H p ( ; Q) is a vector space over Q of dimension β p ( ) Relative homology Suppose is a subcomplex of. C p (, ) is called the group of relative chains of modulo and is defined as C p (, ) = C p ( )/C p ( ) Note that C p (, ) is still a free abelian group. This means that if = S (k 1) = S for some S k then C p (, ) = C p (, S ) C p ( S) where S = k \ S. The boundary operator p : C p ( ) C p 1 ( ) induces a relative boundary operator (see [27] chapter 3), which also is called p or. The relative boundary operator is a homomorphism p : C p (, ) C p 1 (, ) such that = 0. We let Z p (, ) = ker : C p (, ) C p 1 (, ), B p (, ) = im : C p+1 (, ) C p (, ), H p (, ) = Z p (, )/B p (, ). 13

16 These groups are called the group of relative p-cycles, the group of relative p-boundaries and the pth relative homology group of modulo. We see that the relative homology ignores the part of that lies inside and only depends on the part that is on the outside of the subcomplex. Whenever is a subcomplex of there is a long exact sequence H p ( ) H p ( ) H p (, ) H p 1 ( ) (2.1) For a proof, see for example [27] p Cohomology The coboundary operator is the map that corresponds to the transpose of the matrix of the boundary operator. It is the homomorphism p : C p ( ) C p+1 ( ) and for any p-cochain σ we have that p (σ) = ǫ j τ j, where the sum is over all the (p + 1)-cochains whose corresponding (p + 1)-simplex have the p-simplex corresponding to σ as a face. The construction of the cohomology is more complicated than that of homology, see [27] chapter 5, for detailed definitions of for example the group of cochains C p ( ). In this paper we will mainly use that [ p ] = [ p ]. The cocycles are defined as Z p ( ) = ker p, the coboundaries are B p ( ) = im p 1 and finally the cohomology groups are defined as 2.3 Matroids H p ( ) = Z p ( )/B p ( ). A matroid is a structure that generalizes the notion of linear independence from linear algebra. One of many definitions of matroids is that a finite matroid is a pair (E, I) where E is a finite set and I is a simplicial complex on E, called the independent sets, such that if A, B I and A > B then there exists a A \ B such that B {a} I. A basis is a maximal independent set, i.e., if B is a basis then B I and B {e} / I for all e in I \ B. Thus the basis are the facets of I. Any two bases of M have the same number of elements, and hence I is a pure simplicial complex. This number is called the rank of M. A matrix A over a field gives rise to a matroid M = (E, I) where E is the set of columns and the dependent sets I of columns in the matroid are those that are linearly dependent as vectors. A graph G and the set of forests also corresponds to a matroid M = (E, I). Let E be the set of edges, and let the independent sets be the forests, i.e., a subset of the edges is independent if and only if it does not contain a cycle. 14

17 Suppose M = (E, I) is a finite matroid. We can define the dual matroid M by taking the same underlying set E and calling a set a basis in M if and only if its complement is a basis in M. M is a matroid and the dual of M is M. 2.4 Special simplicial complexes A complete graph G = (V, E) is a graph such that every pair of vertices are connected by an edge. Hence a complete graph is such that E = {(v i, v j ) v i, v j V } = P 2 (V ) where P 2 (V ) = {e P(V ) : e = 2}. This idea can be generalized to simplicial complexes. The complete simplicial complex is a simplicial complex, with vertices V, such that = P(V ). A multi-partite (or colorful) graph is a graph G = (V, E) such that V can be divided in disjoint sets, i.e., V = V 1 V m, and there is no edge in E connecting two vertices from the same part. The vertices of V i is said to be of color i and the partition of V into such V 1,..., V m is called a m-coloring of G. This idea too can be generalized to simplicial complex. Let V = V 1... V r be a disjoint union of finite sets. The vertices of V i are said to be of color i. The complete multi-partite (or colorful) simplicial complex on V is defined as = {σ V : σ V i 1, i {1,..., r}}. In a complete multi-partite graph there is an edge connecting any pair of vertices from two distinct parts. A simplicial complex is acyclic if H 0 ( ) = Z and H k ( ) = {0} for all k > 0. Furthermore, is said to be Q-acyclic if H 0 ( ; Q) = Q and H k (, Q) = {0} for all k > 0, which is equivalent to β 0 ( ) = 1 and β k ( ) = 0 for all k > 0. A simplicial complex is collapsible if it can be reduced to a point by a series of elementary collapses. An elementary collapse is the deletion of a facet f of, which has a free face e f, together with the deletion of this free face e. A face is free in a simplicial complex if it is a face of a unique simplex of a larger dimension. So, to shed additional light on this definition, 0 is an elementary collapse of if and only if \ 0 = {f, e} where e f and f = e Notation Let G be a finitely generated abelian group. The torsion of a group G is the subgroup consisting of all the elements g of G such that there exists n N >0 such that ng = 0. The torsion subgroup of G is denoted by [G]. 15

18 Let g 1,..., g n be elements of an abelian group G. Then g 1,..., g n denotes the subgroup of G generated by g 1,..., g n with coefficients from Z. Thus g 1,..., g n equals the subgroup {α 1 g α n g n α 1,..., α n Z}. Suppose p is a natural number. Then [p] = {1, 2,..., p 1, p}. Suppose G is a finite group, then the number of elements in G is denoted by G. Suppose V is a finite set. Then the number of elements in V is denoted by V. P(V ) is the set of all subsets of V, it is also called the power set of V. Let V = n and 0 i n. Then P i (V ) = {E P(V) E = i}. If x 1,..., x n are elements of a Z-module (or a Q-module), then < x 1,..., x n > denotes the submodule generated by x 1,..., x n over Z (or Q). Let [ k ] denote the matrix representation of the map k : C k ( ) C k 1 ( ) with respect to the standard basis in C k ( ) and C k 1 ( ). Furthermore, if M is a matrix and T 1 is a subset of the rows of M and T 2 is a subset of the columns of M then let M T1,T 2 be the submatrix of M with the rows and columns deleted outside T 1 and T 2. If M is a matrix and T is a subset of the columns of M then let M T be the submatrix of M with the columns deleted outside T. Let M denote the transpose of the matrix M. Whenever we have MS,T, apply first the transpose and then the elimination of rows and columns. If S is a subset of k then S = k \ S. 3 Spanning trees in simplicial complexes 3.1 Background In graph theory, a tree is a connected graph without cycles. It is widely used in various applications. For example it is essential for many computer science data structures, such as binary search trees or the Huffman coding, which is an algorithm for lossless data compression. A spanning tree in a graph G = (V, E) is a subgraph that is a tree and has the same vertex set as that of G. A tree is a connected graph without cycles. Spanning trees are useful in many problems concerning electrical networks and Markov chains. From graph theory we have the following theorems: Theorem 3.1 (Kirchoff s Matrix Tree Theorem, 1847). Let G be a graph with n vertices and let T be the set of spanning trees in G. Let L be the 16

19 Laplacian matrix of G and let L i,j be the cofactor of L with the ith row and jth column deleted. We have that T = T T 1 = L i,j for all i, j [n]. Thus the number of spanning trees in G is the determinant of any (n 1) (n 1) submatrix of L. Later Cayley gave a simple formula for the number of spanning trees in the complete graph [6]. Theorem 3.2 (Cayley 1889). Let G be the complete graph with n vertices, K n. Then T = n n 2. Cayley s Theorem was later generalized to the complete r-partite graphs by Austin [3]. Theorem 3.3 (Austin, 1960). Let G be the complete r-partite graph K n1,...,n r. Then r T = n r 2 (n n i ) n i 1 where n = r n i. i=1 i=1 Kalai was the first to generalize this type of theorems concerning graphs to simplicial complexes [18]. He generalized the notion of spanning trees and was also able to obtain Cayley s Theorem for simplicial complexes. Theorem 3.4 (Kalai, 1983 ). Let be the complete simplicial complex of n vertices and let 1 k < n. Let T = T (n, k) be the set of all subcomplexes T of such that (i) T has a complete (k 1)-skeleton, (ii) H k (T) = 0, (iii) H k 1 (T) is finite, then T T H k 1 (T) 2 = n (n 2 k ). 17

20 Thus Kalai introduces the notion of Q-acyclic subcomplexes in complete simplicial complexes. These subcomplexes are exactly the same as the spanning trees in complete simplicial complexes. It is also interesting to note that Bolker, [1] and [4], suggested the right hand side of the formula, but with six vertices it no longer count the number of spanning trees. Kalai s insight that the counting of the spanning trees was weighted with the number of elements in the (k 1)th homology group is fundamental for the subsequent development in this field. Adin [1] continued Kalai s this line of research by generalizing Austins Theorem. He defined spanning trees in complete multi-partite simplicial complexes and proved an enumeration theorem for that case. Theorem 3.5 (Adin 1989). Let V = V 1... V r be a disjoint union of finite sets and let be the complete multi-partite simplicial complex on V. Then a subset T k is called a k-dimensional -tree if and only if (i) H k ( T ) = 0, (ii) H k 1 ( T ) is a finite group. Let T be the family of k-dimensional -trees and let 1 k r 1. Then where H k 1 ( T ) 2 = T T d k ρ (r 2 d k d )π D D D {1,...,r} and d = D ρ D = i/ D n i π D = i D(n i 1) (using the conventions that ( ) m 1 m = 0 for m 0, and an empty product = 1). The number of Adin s spanning trees in a given complete multi-partite simplicial complex depends only on the number of vertices of each color. In the proof of Theorem 3.5, Adin also proved a general Matrix Tree Theorem for complete multi-partite simplicial complexes. 18

21 Theorem 3.6 (Adin s Matrix Tree Theorem). Let V,, T be as in the previous theorem and let L be the Laplacian matrix of. Suppose T k 1 k 1 is a (k 1)-dimensional -tree. Then T T H k 1 ( T ) 2 = det(l T k 1, T k 1 ) H k 2 ( Tk 1 ) 2. Thus, in order to count the number of k-dimensional -trees, with weights, Theorem 3.6 requires us to find a (k 1)-dimensional -tree. 3.2 Definition of spanning trees based on homology There are many ways to generalize the concept of trees in graph theory to the theory of simplicial complexes. Those made by Kalai, Adin, Klivans, Duval, Martin and Lyons and the ones presented in this paper are all consistent. In this chapter we will first study the above mentioned definitions and, for the sake of completeness, we will also mention some definitions which do not coincide with those mentioned above. Kalai defines spanning trees when the original complex is the complete simplicial complex on a vertex set V of size n. He does not refer to them as trees but uses the notation C(n, k). This definition is slightly reformulated compared to the definition Kalai gives, see [18]. Definition 3.7 (Spanning trees according to Kalai). Let be the complete k-dimensional complex on n vertices. A k-dimensional simplicial subcomplex C of with a complete k 1-skeleton is a k-dimensional spanning tree of if (i) C k = ( ) n 1 k, (ii) H k (C) = 0, (iii) H k 1 (C) is a finite group. Adin uses the term tree and defines spanning trees when is the complete colorful (or multi-partite) simplicial complex on a vertex set V = V 1... V r, where denotes disjoint union. Definition 3.8 (A k-dimensional -tree according to Adin). Suppose is a complete colorful simplicial complex of dimension d and 0 < k d. A k-dimensional -tree is a subset T k such that (i) H k ( T ) = 0, 19

22 (ii) H k 1 ( T ) is a finite group. Duval, Klivans and Martins defines spanning trees only in simplicial complexes such that β i ( ) = 0 for all i < k. In their article [7], Duval, Klivans and Martin point out that a graph G has a spanning tree if and only if G is connected i.e. if and only if H i (G; Q) = {0}. That is, when i is less than the dimension of the simplicial complex, then H i (G; Q) = {0}. Therefore a corresponding condition for a k-dimensional simplicial complex would be that H i ( ; Q) = {0} if i < k, where k is the dimension of the simplicial complex. Definition 3.9 (Spanning trees according to Duval, Klivans and Martins). Let be a d-dimensional simplicial complex and suppose is (d 1)-Qacyclic. Let 0 < k d, then a k-dimensional simplicial spanning tree of is a k-dimensional subcomplex Υ such that (i) Υ (k 1) = (k 1), (ii) H k (Υ) = 0, (iii) β k 1 (Υ) = 0, (iv) Υ k = k β k ( ) + β k 1 ( ) Definition in the special case β k 1 ( ) = 0 Suppose is a pure and connected simplicial complex of dimension d. In this case a spanning tree of is defined as follows. Definition Suppose 0 < k d. A k-dimensional -tree when is such that β k 1 ( ) = 0 is a subset T such that (i) H k ( T ) = {0}, (ii) β k 1 ( T ) = 0. Note that, since β k 1 ( ) = 0 we have that β k 1 ( ) = β k 1 ( T ) Spanning trees defined for any pure simplicial complex In graph theory a spanning tree τ in a graph G = (V, E) is a spanning subgraph, i.e., τ = (V, E ) without cycles. Let be a pure r-dimensional connected simplicial complex. will play the role of G in the previous example with graphs. 20

23 Example 3.11 (Holes). Consider the simplicial complex seen in Figure 3.1 What would be a 2-dimensional -tree in this case? The first of Adin s conditions, H 2 ( T ) = {0}, is fulfilled by all the 2-dimensional subcomplexes of. But the second condition, β 1 ( T ) = 0, is not fulfilled by any subset T of the 2-faces of. Figure 3.1: The simplicial complex in Example The colored triangles correspond to a 2-dimensional simplex in the simplicial complex. β 0 ( ) = 0, β 1 ( ) = 1, β 2 ( ) =. It is the structure of that makes it impossible with β 1 ( T ) = 0. Instead the condition should be β 1 ( T ) = β 1 ( ). But could this give us -trees with holes larger than? No, since T contains 1, we would then get β 1 ( T ) > β 1 ( ). Example 3.12 illustrates this fact. Example Let be the simplicial complex in Figure 3.2. The subset T 2 and the subcomplex T is seen in Figure 3.3. Then H 2 ( T ) = {0}, but β 1 ( T ) = 2 1 = β 1 ( ). Figure 3.2: in Example All the colored triangles correspond to a 2-dimensional simplex in the simplicial complex. The starting point was that every pure and connected simplicial complex should have a spanning tree. This idea was supported by the results of experimental studies. Our definition of spanning tree is the following: 21

24 Figure 3.3: Complexes in Example To the left we have the facets in T. Every colored triangle corresponds to a three point set in T. The simplicial complex to the right is T, where every colored triangle corresponds to a 2- dimensional simplex and every edge corresponds to a 1-dimensional simplex in T. Definition 3.13 (k-dimensional -tree). A k-dimensional -tree, or a spanning k-dimensional tree of, is a subset T k such that (i) H k ( T ) = 0, (ii) β k 1 ( T ) = β k 1 ( ). Note that this definition implies that β l ( T ) = β l ( ) for all 0 l k 1. This is because we have partly condition (ii), and partly that β l ( T ) = β l ( ) for all 0 l k 2, since T and share the same (k 1)-skeleton. This definition gives, as we will see, a spanning tree to every pure and connected simplicial complex. Occasionally we will consider trees as well, and not just spanning trees. Definition 3.14 (k-tree). Let C be a pure k-dimensional connected simplicial complex. If C is such that H k (C) = 0 then we call C a k-tree or a k-dimensional tree. Note that the 1-trees are exactly the same object as the trees in graph theory. That is, a tree is a connected graph without cycles. Notation Let T k ( ) denote the family of k-dimensional spanning trees in. Lyons also worked on the same problem but from a different angle. He defines a k-base of as the following. 22

25 Definition 3.16 (k-base of ). Suppose T k. Then T is a k-base of if it is a base of the matroid defined by the matrix [ k ]. An equivalent condition is that T is maximal with the property Z k ( T ) = {0}. Note that a k-base is a set T k which is maximal in this particular regard: { (f)} f T are linearly independent (where is the usual boundary operator). We will see that T a spanning tree in is equivalent to T a k-base of. 3.3 Definition of (k 1)-cobases In order to generalize Adin s Matrix Tree Theorem we also need a (k 1)- dimensional subcomplex to play the role of T k 1 in Adin s Theorem. But the following example shows that in our case this subcomplex is not going to be a (k 1)-dimensional -tree. Therefore this subcomplex is called S instead of T k 1 from now on. Example Let be the simplicial complex in Figure 3.4 whose underlying space is the Klein bottle. Then H 2 ( ) {0}, H 1 ( ) Z Z 2 and H 0 ( ) {0}. Let T be a 2-dimensional -tree. Then H 2 ( T ) {0} and β 1 ( T ) = 1. From Proposition 5.2 we know that T = k β k ( ) = 20 0 = 20. Hence the only 2-dimensional spanning tree in is itself. Furthermore, S = 20 and S = 1 \ S = = 10. Since V = 10 we know that S must contain exactly one cycle (it is a graph with as many edges as vertices), that is H 1 ( S ) Z {0}. a b c a d e e d a b c a Figure 3.4: The simplicial complex in Example The underlying space of is the Klein bottle. Every basic triangle (i.e. triangle that do not consist of smaller triangles) correspond to a 2-dimensional simplex in the simplicial complex. In the experiments many choices of S do not work (that is, the determinant is zero even if we have a tree) even though they have the right cardinality. We need further conditions on S. 23

26 But will two choices of S, say S A and S B give the same result if H 1 ( SA ) = H 1 ( SB )? The answer is no, as is illustrated by Example Example Let be the simplicial complex in Figure 3.4 whose underlying space is the Klein bottle and let S A and S B be the subsets of 1 shown in Figures 3.5 and 3.6, respectively. We have that det(l S A, S A ) = 4 and det(l SB, S B ) = 16. Note that H 1 (, SA ) Z 2 and H 1 (, SB ) Z 2 Z 2. In this case there is no S such that H 1 (, S ) = {0}, so we cannot use that as a condition on S. But H 1 (, S ) is finite in both cases. That is equivalent to β 1 (, S ) = 0. a b c a d e e d a b c a Figure 3.5: The thicker edges are the edges in S A. a b c a d e e d a b c a Figure 3.6: The thicker edges are the edges in S B. 3.4 Experimental methods Our starting point was to generalize the Matrix Tree Theorem where the absolute value of the determinant of any cofactor of the Laplacian matrix of the graph G equals the total number of spanning trees in G. Since a graph also can be seen as a simplicial complex and since the Laplacian matrix is defined for simplicial complexes in general, the idea was to look for a 24

27 definition of spanning trees in simplicial complex such that the determinant of a submatrix of the Laplacian matrix for would be related to the number of spanning trees in. Therefore we started with computer experiments where a lot of Laplacian matrices corresponding to simplicial complexes of different types and their subdeterminants were examined. The experiments where restricted to 2-dimensional and pure simplicial complexes. The procedure was to construct the matrix [ ] corresponding to the boundary operator of the original simplicial complex. Then the rank of the matrix [ ] was calculated, call it g, and subsequently the determinant of every submatrix of size g g of [ ] was calculated. Finally a 1-cobase S was chosen and the determinant of the submatrix of the Laplacian matrix L S,S was computed and the number compared with the number of spanning trees counted by hand. This procedure was first applied to simplicial complexes such that H 1 ( ) {0} and every thing worked out well, i.e., when the rows of [ ] constituted a 1-cobase the determinants where either 0, and then the 2- simplices corresponding to the chosen columns was not a spanning tree of, or ±1 if these columns did correspond to a spanning tree. For example every simplicial complex on five vertices or less was examined. The complete simplicial complex on five vertices has 125 spanning trees. The complete simplicial complex on six nodes has, however, spanning trees [18], so the number of possible simplicial complexes is already at six nodes huge. The experiments therefore continued on simplicial complexes such that H 1 ( ) {0}, for example simplicial complexes whose underlying space is the Klein bottle, RP 2 or the Dunce Hat. The spanning trees and their appearance and homological properties where studied along with the appearance and homological properties of the choice of rows, which corresponds to the cobases, as we will see later. The determinant of the Laplacian corresponding to the Klein bottle was ±4 or ±16, depending on the choice of S even though there was only one spanning tree, i.e, the whole simplicial complex. The result for RP 2 was ±4 and in this case to, there was only one spanning tree. The explanation was that the enumeration was weighted with the square of the number of elements in the torsion subgroup of each tree, that is, weighted with the number [H 1 ( T )] 2. Remember that the first homology group for the Klein bottle is Z Z 2 and for the real projective plane it is Z 2. 25

28 4 Alternative notions of trees in simplicial complexes In an attempt to give a broader perspective on the subject of trees in simplicial complexes we have gathered some other definitions of trees in simplicial complexes and hypergraphs which generalizes other properties of trees in graph theory. Since the HP-trees, MV -trees and F-trees are not compatible with the trees presented in this paper, see Definition 3.14, the definitions needed are presented separately in each section. Since all definitions starts with concepts from graph theory, many definitions had to be given different prefixes (the initials of the corresponding inventors) to avoid multiple definitions of the same word. The reader can continue to Section 5 without any loss of understanding. Remember that k-trees are the k-dimensional simplicial complexes τ that are pure, connected and such that H k (τ) = {0}. In this paper the trees defined by Harary and Palmer are called HP-trees and the trees defined by Masbaum and Vaintraub are called MV -trees, to avoid confusion. 4.1 The trees of Harary and Palmer In their article [14] Harary and Palmer discuss only 2-dimensional trees. For convenience 0-simplices, 1-simplices and 2-simplices are referred to as vertices, edges and triangles, respectively. In a given 2-dimensional simplicial complex, an HP-walk is an alternating sequence e 0, f 0, e 1, f 1,..., e n 1, f n 1, e n of edges e i and triangles f i such that each edge e i is incident with the triangles f i 1 and f i. This walk is called HP-closed if e 0 = e n and is called HP-open otherwise. If all triangles are distinct it is an HP-trail and if both all edges and all triangles are distinct it is an HP-path. We say that a simplicial complex is HP-connected when there is a 2-path connecting any pair of its edges. It is called HP-acyclic if it contains no closed 2-trail with more than one cell. Furthermore it is HP-simply connected if it is connected and its first Betti number (β 1 ) is zero. An HP-tree is a 2-dimensional simplicial complex that is both HP-simply connected and HP-acyclic. 26

29 Figure 4.1: This simplicial complex is a 2-dimensional tree but not an HPtree since it is not HP-connected Figure 4.2: This simplicial complex is a 2-dimensional tree and it is HPconnected but still not an HP-tree since it is not HP-simply connected 4.2 The trees of Masbaum and Vaintrob In their paper [26] Masbaum and Vaintrob gives another definition of trees in simplicial complexes which is not consistent with either HP-trees or the trees in Definition Although the MV -trees looks quite different from the trees by our notion, they supply an enumeration theorem based on pfaffians, which are related to determinants. They only discuss 2-dimensional simplicial complexes and they choose to regard them as 3-graphs, that is hypergraphs where the edges has three vertex points. Remember that a hypergraph H is a pair H = (V, E) where V is a set of elements, called vertices, and E is a set of non-empty subsets of V called hyperedges. In this case, since we only consider 3-graphs, we have that e = 3 for all e E. An MV-tree is a 3-graph where the underlying graph is a tree in the usual graph theoretical sense. The underlying graph is obtained by substituting every hyperedge {v 1, v 2, v 3 } with the Y-shaped graph G = ({v 1, v 2, v 3, v 4 }, {{v 1, v 4 }, {v 2, v 4 }, {v 3, v 4 }}) 27

30 Figure 4.3: This simplicial complex is a 2-dimensional tree and it is HPconnected and HP-simply connected but still not an HP-tree since it is not HP-acyclic Figure 4.4: This simplicial complex is a 2-dimensional tree and an HP-tree where v 4 V, see Figure 4.5. In the following figures the underlying graph will be denoted in the same figure as the simplicial complex itself. A spanning MV-tree in a 3-graph G = (V, E) is an MV -tree whose edges contain all the vertices of G. This definition implies that many 3-graphs do not have a spanning tree, see for example Figure 4.7. In fact, all MV -trees V 1 has an odd number of vertices and E =. Hence all 3-graphs that do 2 have a spanning MV -tree must have an odd number of vertices as well Enumeration theorem for spanning MV -trees In their paper, Masbaum and Vaintrob also gives an enumeration theorem for the number of spanning trees in a 3-graph, it is called the Pfaffian Matrix Tree-Theorem. Let G be a 3-graph with vertices numbered from 1 to m. The Pfaffian-tree 28

31 Figure 4.5: A 2-dimensional simplicial complex, or a 3-graph, to the left and its underlying graph to the right. Note that this simplicial complex is an MV -tree and a tree but not an HP-tree. Figure 4.6: A 2-dimensional simplicial complex, with its underlying graph indicated by dotted lines. Note that this simplicial complex is an HP-tree but not an MV -tree. polynomial of G is P G = ( 1) p 1 Pf(Λ(G) (p) ), where Λ(G) (p) for p = 1,..., m is the matrix obtained by removing the pth row and column from Λ(G). Λ(G) is the skew-symmetric matrix defined by Λ(G) = (λ ij ) 1 i,j m where λ ii = 0 for all i {1,..., m} and λ ij = yẽ where ẽ ẽ runs through the oriented edges Ẽ of G such that i and j are vertices of ẽ, and the orientation of ẽ is represented by the cyclic ordering (ijk), where k denotes the third vertex of ẽ. Ẽ is the set of oriented edges of G. To every oriented edge ẽ Ẽ we associate an indeterminate y ẽ with the relation y ẽ = yẽ where ẽ denotes the edge e with the opposite direction compared to ẽ. A Pfaffian Pf(A) of a matrix A is non-vanishing only for 2n 2n skewsymmetric matrices in which case it is equal to a polynomial of degree n. One can associate to any skew-symmetric 2n 2n matrix A = [a i,j ] a bivector ω = i<j a ije i e j where e 1,..., e 2n is the standard basis of R 2n. The Pfaffian is then defined by the equation 1 n! ωn = Pf(A)e 1 e 2n, where ω n denotes the wedge product of n copies of ω with itself. 29

32 Figure 4.7: A 2-dimensional simplicial complex, with its underlying graph indicated by dotted lines. It is a tree, but neither an HP-tree nor an MV -tree. Figure 4.8: A 2-dimensional simplicial complex, with its underlying graph indicated by dotted lines. It is a tree and an MV -tree but not a HP-tree since it is not HP-connected. Furthermore, let G be a 3-graph, which is an MV -tree, and let o be an orientation of T. We define the monomial y(t, o) by the formula y(t, o) = o( T) o e E(T) yẽ where T is the tree with any choice of orientations for the edges, o(t) = o( T) e E(T) σ(e) and is the sign relating the two orientations. o Let G be a 3-graph and let o be an orientation of G. The generating function for spanning trees of G is P(G, o) = T y(t, o) where T runs through the spanning trees of G. 30

33 Theorem 4.1 (The Pfaffian-Tree Matrix Theorem). Let G be a 3-graph with vertices numbered from 1 to m. Then the generating function of spanning trees in G is equal to the Pfaffian-tree polynomial of G, P(G, o can ) = P G where o can is the canonical orientation of G determined by the ordering of the vertices. For the proof and more details regarding definitions, we refer the reader to [26]. 4.3 The trees of Faridi Faridi has been working on trees in simplicial complexes as well and has published several papers in the field, see for example [9], [10] and [11]. The trees of Faridi will be named F-trees in this paper. F-trees generalizes the property of graph-theoretical trees that every non-empty subgraph of a tree has a leaf, where a leaf is a vertex that belongs to a unique edge. Let F 1,..., F n be the set of facets of a simplicial complex. F i is called a leaf of if either F i is the only facet of, or if P(F i ) i j P(F j ) is a face of. Note that a leaf must have at least one free vertex. A connected simplicial complex is an F-tree if the simplicial complex induced by every subcollection of facets of has a leaf. Note that F-trees do not need to be pure. Faridi shows, among other things, that the F-trees are sequentially Cohen Macaulay, [10], and gives an algorithm that checks in polynomial time whether a simplicial complex is an F-tree, [11]. 5 Some lemmas valid for any simplicial complex In this section we present some lemmas and propositions that are useful both in the special and the general case. These results are valid for any pure and connected d-dimensional simplicial complex. Several of the arguments in this section essentially follow the same lines as Adin s proof of the Matrix Tree Theorem for complete colorful simplicial complexes. 31