Logarithmic Tree-numbers for Acyclic Complexes

For a d-dimensional cell complex Γ with˜H i (Γ) = 0 for −1 i < d, an i-dimensional tree is a non-empty collection B of i-dimensional cells in Γ such that˜H i (B ∪ Γ (i−1)) = 0 and w(B) := | ˜ H i−1 (B ∪ Γ (i−1))| is finite, where Γ (i) is the i-skeleton of Γ. The i-th tree-number is defined k i := B w(B) 2 , where the sum is over all i-dimensional trees. In this paper, we will show that if Γ is acyclic and k i > 0 for −1 i d, then k i and the combinatorial Laplace operators ∆ i are related by d i=−1 ω i x i+1 = (1 + x) 2 d−1 i=0 κ i x i , where ω i = log det ∆ i and κ i = log k i. We will discuss various consequences and applications of this equation.


Introduction
In this paper, we will extend Temperley's tree-number formula for finite graphs [13] to a class of cell complexes, called γ-complexes, and show applications to various acyclic complexes.
As the main object of study in this paper, we define a γ-complex to be a non-empty finite cell complex Γ whose integral cellular chain complex {C i , ∂ i } with C −1 = Z satisfies the following conditions: (γ1) ∂ i = 0 for 0 i dim Γ, and (γ2) the reduced integral homology Hi (Γ) = 0 for i < dim Γ.This definition is intended to be a generalization of connected finite graphs.Other examples of γ-complexes are matroid complexes, standard simplexes, and cubical complexes [4] with the latter two being acyclic.Note that a γ-complex is a special case of APC (acyclic in positive codimension) complexes in the terminology of [4].
We define high-dimensional spanning trees for a γ-complex extending the ideas in [1].Given a γ-complex Γ, let Γ i be the set of all i-dimensional cells, and Γ (i) the i-skeleton of Γ.Given a subset S ⊂ Γ i , define Γ S = S ∪ Γ (i−1) as a subcomplex of Γ.An idimensional spanning tree of Γ (or simply, an i-tree) is a non-empty subset B ⊂ Γ i such that Hi (Γ B ) = 0 and w(B) := | Hi−1 (Γ B )| is finite.Define the i-th tree-number of Γ by where the sum is over all i-trees in Γ.We will see that k i > 0 for all −1 i dim Γ where we define k −1 = 1.If Γ is a graph, then k 0 is the number of vertices and k 1 is the number of spanning trees in Γ.
An important method for computing the tree-numbers for Γ is given by the combinatorial Laplacians ∆ i ( [1], [4], and [13]).For example, let ∆ 0 = L + J, where L is the Laplacian matrix of a finite graph G of order n, and J is the all 1's matrix.Temperley [13] showed that det ∆ 0 = n 2 k 1 for G (refer to Theorem 4).This method is more efficient than the matrix-tree theorem for certain graphs.Indeed, for Γ = K n the complete graph on n vertices, we have ∆ 0 = nI and det ∆ 0 = n n , from which the Cayley's Theorem We will show that Temperley's formula can be extended to any γ-complex Γ (refer to Proposition 7).Also, if Γ is acyclic of dimension d, then each ∆ i is positive-definite, and the following polynomials are well-defined: The main result of the paper is A refinement of this equation and its applicability to matroid complexes will be discussed through a simple example.(See Section 5.) This paper is organized as follows.Section 2 is a review of useful facts from matrix theory and combinatorial Laplacians for γ-complexes.It also provides a proof of Temperley's tree-number formula.In Section 3, we will describe high-dimensional spanning trees for a γ-complex via the boundary operators of its chain complex.In Section 4, we will prove the main results of the paper which consist of a generalization of Temperley's tree-number formula and a logarithmic version (1) of this result for acyclic γ-complexes.In Section 5, we will discuss applications of (1) to standard simplexes [7], the cubical complexes [4], and an example of graphic matroid complex.

Lemmas from Matrix Theory
We will review several important facts about symmetric matrices.For definitions and basic facts from matrix theory, one may refer to [6].All matrices are assumed to have real entries.For a square matrix M , let P M denote the multiset of all non-zero eigenvalues of M , and let π M = λ∈P M λ .The following two lemmas and their proofs appear in [1].We will sketch the proofs here.Lemma 1.Let A and B be n × n symmetric matrices such that AB = BA = 0.Then, P A+B = P A ∪ P B as a multiset.In particular, if A + B is non-singular, Proof.Since A and B are symmetric and they commute, they are simultaneously diagonalizable.For each i ∈ [1, n], let λ i and µ i be the eigenvalues of A and B, respectively, so that the collection } is the multiset of all eigenvalues of A + B. Since AB = 0, we have either λ i = 0 or µ i = 0 for each i.
Lemma 2. Let M be a rectangular matrix of rank r (r > 0).Let B(M ) be the collection of all non-singular r×r submatrices of Proof.This result follows from Binet-Cauchy theorem and the fact that the product of all non-zero eigenvalues of a diagonalizable matrix of rank r equals the sum of all principal minors of order r.Equation (3) holds for both M M t and M t M because they have the same multiset of non-zero eigenvalues.Details will be omitted.

Combinatorial Laplacians for γ-complexes
We will assume familiarity with basic definitions concerning finite cell complexes and reduced homology groups.One may refer to standard texts such as [10] for details.Let X be a finite cell complex of dimension d.
Since our main object of study is a γ-complex, we will consider only those X such that X i = ∅ for all i ∈ [0, d].This condition on X allows one to represent the boundary maps ∂ i of its chain complex as matrices.Also we define X −1 to be a set with one element.
For i ∈ [−1, d], the i-th chain group of X is the free abelian group C i ∼ = Z |X i | generated by X i .Let {C i , ∂ i } be an augmented chain complex of X with the augmentation ∂ 0 : C 0 → C −1 ∼ = Z given by ∂ 0 (v) = 1 for every v ∈ X 0 .The i-th reduced homology group of X is defined Hi (X) = Ker ∂ i /Im ∂ i+1 where we define ∂ d+1 and ∂ −1 to be zero maps.Hence, we have Hd ( whose rows and columns are indexed by X i−1 and X i , respectively.In particular, the augmentation ∂ 0 is an all 1's row matrix of size |X 0 |.The coboundary map ∂ t i : Hence, each ∆ i is also symmetric and non-negative definite by Lemma 1.
An important property of ∆ i is that the dimension of the 0-eigenspace of ∆ i as an operator on a finite dimensional vector space over Q equals the dimension of the reduced rational homology Hi (X; Note that ∆ −1 = L −1 : Z → Z is a multiplication by |X 0 |.Now the following lemma is immediate from the definition of γ-complex and (4).

Temperley's tree-number formula
For a finite loopless graph G with n vertices and its Laplacian matrix L(G), Temperley [13] showed the following analogue of the Matrix-Tree theorem [8] for the number of spanning trees k(G) in G. Let J denote the all 1's matrix.
Proof.We will give a proof of this formula as a consequence of the multilinearity of determinant function and the Matrix-Tree theorem.We refer the readers to [2] for a proof via eigenvalues.
Let L(G) where C i 's and D i 's are the columns of L(G) and J, respectively.Given a subset S ⊂ [n], define M S = (X 1 , X 2 , . . ., X n ), where By the multilinearity of determinant function (on columns), det(L(G) + J) = S⊂[n] det M S , where the sum is over all subsets S of [n].
Clearly, we have det 0 because rank of J is 1.However, for every i ∈ [n], we see that det M {i} = nk(G) because every entry in D i is 1 and every cofactor of L(G) equals k(G) by the Matrix-Tree theorem.Therefore, we have the electronic journal of combinatorics 21(1) (2014), #P1.50 We make two observations about Theorem 4. First, unlike Matrix-Tree theorem, Temperley's formula does not require deletion of a row and a column from L(G) to compute k(G).Second, regarding G as a 1-dimensional γ-complex, one can check that As we shall see, similar observations can be made in computing high-dimensional treenumbers for γ-complexes using combinatorial Laplacians.In particular, one can easily check that equation ( 5) is a consequence of Proposition 7.

High-dimensional trees for γ-complexes
We refer the readers to [1], [4], and [7] for details of high-dimensional trees and of the exact homology sequence used in the proof of Theorem 6.In this section, Γ will denote a γ-complex of dimension d.For a non-empty subset S ⊂ Γ i , define Γ S := S ∪ Γ (i−1) Note that condition 3 is a consequence of the fact Γ (i−1) B = Γ (i−1) .We will denote the set of all i-trees in Γ by B i = B i (Γ) with B −1 = {∅}.It is clear that B 0 is the set of all single 0-cells in Γ and B 1 is the set of all graph theoretic spanning trees of Γ (1) as a finite graph.
Define the i-th tree-number of Γ to be We have k −1 = 1 by definition, and k 0 = |Γ 0 |.If Γ is a connected graph, then k 1 is the number of spanning trees in Γ because w(B) = 1 for B ∈ B 1 .However, w(B) may not equal 1 for B ∈ B i when i > 1. (See [7].)Next, we will describe i-trees via the boundary operator ∂ i of Γ, which will show that (Refer to [2] for details.)More generally, we have the following useful fact.
Then B i is non-empty, and it is given by = Γ (i−1) and Hi−1 (Γ B ) is finite, we must have rk ∂ B = n i−1 the rank of Ker ∂ i−1 .However, Hi−1 (Γ) = 0 implies r i = n i−1 , and we have |B| = rk ∂ B = r i .The inclusion of the right-hand side of (6) in B i is proved similarly.The second statement follows from Remarks 1.In matroid theoretic terms, B i is the set of all bases of a matroid whose ground set is Γ i and the independent sets are the subsets I ⊂ Γ i such that Ker ∂ I = 0 or I = ∅.(Refer to [11] for the definition of a matroid.) 2. If Γ is also acyclic, then there is exactly one d-tree, namely B = Γ d .Since Ker ∂ d = Hd (Γ) = 0, the only base of the matroid just mentioned is Γ d .In this case, it also follows that k d = 1 because Hd−1 (Γ B ) = Hd−1 (Γ) = 0. 3.If X is a cell complex satisfying (γ2) but r i = 0 for some i, then X has no i-tree.Indeed, for any non-empty subset S ⊂ Γ i , we would have Hi (Γ S ) = Z |S| = 0.
The following theorem will play an essential role in Section 4. Given non-empty subsets S ⊂ Γ i−1 and T ⊂ Γ i , let ∂ S,T be the |S| × |T | submatrix of ∂ i whose rows and columns are indexed by S and T , respectively.Denote S = Γ i−1 \ S.
Then the set of all r i ×r i non-singular submatrices of ∂ i is given by Proof.Let S ⊂ Γ i−1 with |S| = r i−1 and let T ⊂ Γ i with |T | = r i .Then ∂ S,T is a square submatrix of ∂ i of order r i by Prop. 5. First, we will show that ∂ S,T is singular if S / ∈ B i−1 or T / ∈ B i .Regard ∂ S,T as the top boundary operator for the relative complex (Γ T , Γ S ).Note that Hi (Γ T ) = Ker ∂ T , Hi (Γ T , Γ S ) = Ker ∂ S,T , and Hi−1 (Γ S ) = Ker ∂ S .Since Hi (Γ S ) = 0, we obtain the following exact sequence from the long exact homology sequence of the pair (Γ T , Γ S ) :

If T /
∈ B i , then Ker ∂ T = 0 by Remark 1 above.Hence, we have Ker ∂ S,T = 0. Similarly, if S / ∈ B i−1 , then rk(Ker ∂ S ) = 0.If T / ∈ B i , we are done.If T ∈ B i , then Ker ∂ T = 0 and Hi−1 (Γ T ) is finite.Therefore, it is clear that Ker ∂ S,T = 0. Now we proceed to prove the second statement, which will also complete the proof of the first statement.Consider the following portion of the long exact homology sequence of the pair (Γ B , Γ A ) with A ∈ B i−1 and B ∈ B i :

Main Results
The following proposition is a generalization of Temperley's tree-number formula (5) for γ-complexes.
Proposition 7. Let Γ be a γ-complex of dimension d, and let ∆ i be its combinatorial Laplacians for i (2) Note that we have rk Therefore, ∂ i ∂ t i has non-zero eigenvalues.Let π i denote the product of all non-zero eigenvalues of ∂ i ∂ t i .By Lemma 2 and Theorem 6, we have have the same multiset of non-zero eigenvalues.Therefore, for i ∈ [0, d−1], Lemma 1 and Lemma 3 imply If Γ is not acyclic, then rk Hd (Γ) > 0 and det ∆ d = 0 by (4).
As we discussed relations between Theorem 4 and the Matrix-Tree theorem in Section 2.3, we can make similar observations about Proposition 7 and the Cellular Matrix-Tree Theorem [4, Theorem 2.8] as follows.Proposition 7 may be derived from the Cellular Matrix-Tree Theorem for general APC complexes X, which states that π as in the above proof.Also, if X is an acyclic γ-complex, then the following theorem, a logarithmic version of Proposition 7 for acyclic γ-complexes, shows that highdimensional tree numbers of X can be obtained without using reduced Laplacians.Refer to Section 5 for examples.
, where ω i = log det ∆ i and κ i = log k i .Then we have Proof.Since Γ is a γ-complex, we have B i = ∅ and k i Hence K(x) is well defined.By Proposition 7, we see that det ∆ i 1 for i ∈ [−1, d], and D(x) is well defined.The rest of the proof is checking the following details.Proposition 7 (1) implies which also follows from (5).Also, k d = 1 because Γ is acyclic, and we have The result follows.
The requirement that Γ be a γ-complex is important in Theorem 8.For example, one can construct an acyclic cell complex consisting of one 0-cell, one 2-cell, and one 3-cell which is not a γ-complex because ∂ 1 = ∂ 2 = 0 in its cellular chain complex.In this case, ∆ 1 = 0 and the above theorem cannot be applied.
As a corollary to Theorem 8, we obtain the following interesting property of the combinatorial Laplacians for acyclic γ-complexes.Refer to [12] for further discussions.
Remarks.Theorem 8 can be refined as follows.Let det ∆ i = p p p,i be the prime decomposition of the positive integer det ∆ i .Let P be the set of all distinct primes that appear in these prime decompositions.For each p ∈ P, define Then, D(x) = p∈P D p (x).Also, we claim that each D p (x) is divisible by (1 + x) 2 .Indeed, suppose D p (x) ≡ log p (a p x + b p ) mod (1 + x) 2 for some integers a p and b p .Since D(x) ≡ 0 mod (1 + x) 2 , we must have p∈P log p (a p x + b p ) = 0. From this equation, one can show that a p = 0 and b p = 0 for each p ∈ P. See Section 5.3 for an example.

Standard simplexes
Let Σ be the standard simplex on n vertices (hence dim Σ = n − 1).Σ is acyclic and ∂ i and ∂ i+1 (and their transpose).Therefore, we have ∆ i = nI, where I is the identity matrix of order n i+1 , and det ∆ i = n ( n i+1 ) .Letting ω i = log n det ∆ i = n i+1 , we see that By Theorem 8, we obtain where . Hence, we have . This result was originally obtained by Kalai [7].

Cubical complexes
The n-cube Q n (n 1) is an n-dimensional cell complex that is the n-fold product I × • • • × I, where I is the unit interval regarded as a cell complex with two 0-cells and one 1-cell.Hence Q n is a cell complex of dimension n, and is the convex hull of the 2 n points in R n whose coordinates are all 0 or 1.One can see that Q n is acyclic by induction on n together with the fact that Q n−1 is a deformation retract of Q n for n 2.
In [4], Duval, Klivans, and Martin showed that the tree-numbers for Q n are based on the spectra (the multisets of eigenvalues) of ∂ i ∂ t i , which are, in turn, obtained from those of ∆ i 's.In what follows, we will derive (8) directly from the spectra Spec(∆ i ) of ∆ i via Theorem 8. We will start with the following generating function for the eigenvalues of ∆ i 's for Q n ([4, Theorem 3.4]): where . By Theorem 4, we also obtain det ∆ 0 = 2 n n j=1 (2j) ( n j ) .Now, let ω i = log 2 det ∆ i , and let α j = n j log 2 (2j).Then, the electronic journal of combinatorics 21(1) (2014), #P1.50 and we have By Theorem 8, we obtain where κ i = log 2 k i .By identifying the coefficients of

A non-acyclic example
Let X be a 2-dimensional simplicial complex on the vertex set E = {a, b, c, d, e} given by , and X 2 = E 3 \ {{a, b, e}, {c, d, e}}.One can check that X is the independent set complex of a cycle matroid of the graph K 4 \ {an edge}.(Refer to [3] for general matroid complexes.)In particular, X has the homotopy type of a bouquet of two-dimensional spheres.Hence it is a γ-complex of dimension 2.
For convenience, assume that simplices in each X i (i = 0, 1, 2) are ordered lexicographically, and that the alphabetical ordering of vertices in each simplex gives the positive orientation for the corresponding oriented simplex.For example, C 2 for X is isomorphic With this new "augmented" acyclic complex, one can show det ∆ −1 = 5, det ∆ 0 = 5 5 , det ∆ 1 = 2 2 5 8 , det ∆ 2 = 2 4 5 5 , and det ∆ 3 = 2 2 5 .By Proposition 7, det ∆ 2 and det ∆ 3 are independent of the choices of z 1 and z 2 because k 2 depends only on ∂ 2 and k 3 = 1.The primes that appear in these prime decompositions are P = {2, 5}, and we see that Question.Can one characterize tree numbers of a matroid complex via known matroid invariants?