Simplicial Matrix-Tree Theorems

Simplicial Matrix-Tree Theorems

Art M. Duval Department of Mathematical Sciences, University of Texas at El Paso, El Paso, TX 79968-0514 Caroline J. Klivans Departments of Mathematics and Computer Science, The University of Chicago, Chicago, IL 60637  and  Jeremy L. Martin Department of Mathematics, University of Kansas, Lawrence, KS 66047
August 14, 2008

We generalize the definition and enumeration of spanning trees from the setting of graphs to that of arbitrary-dimensional simplicial complexes , extending an idea due to G. Kalai. We prove a simplicial version of the Matrix-Tree Theorem that counts simplicial spanning trees, weighted by the squares of the orders of their top-dimensional integral homology groups, in terms of the Laplacian matrix of . As in the graphic case, one can obtain a more finely weighted generating function for simplicial spanning trees by assigning an indeterminate to each vertex of and replacing the entries of the Laplacian with Laurent monomials. When is a shifted complex, we give a combinatorial interpretation of the eigenvalues of its weighted Laplacian and prove that they determine its set of faces uniquely, generalizing known results about threshold graphs and unweighted Laplacian eigenvalues of shifted complexes.

Key words and phrases:
Simplicial complex, spanning tree, tree enumeration, Laplacian, spectra, eigenvalues, shifted complex
2000 Mathematics Subject Classification:
Primary 05A15; Secondary 05E99, 05C05, 05C50, 15A18, 57M15
Second author partially supported by NSF VIGRE grant DMS-0502215. Third author partially supported by an NSA Young Investigators Grant.

1. Introduction

This article is about generalizing the Matrix-Tree Theorem from graphs to simplicial complexes.

1.1. The classical Matrix-Tree Theorem

We begin by reviewing the classical case; for a more detailed treatment, see, e.g., [8]. Let be a finite, simple, undirected graph with vertices and edges . A spanning subgraph of is a graph with and ; thus a spanning subgraph may be specified by its edge set. A spanning subgraph  is a spanning tree if (a) is acyclic; (b) is connected; and (c) . It is a fundamental property of spanning trees (the “two-out-of-three theorem”) that any two of these three conditions together imply the third.

The Laplacian of is the symmetric matrix with entries

where is the degree of vertex  (the number of edges having as an endpoint). Equivalently, , where is the (signed) vertex-edge incidence matrix and is its transpose. If we regard as a one-dimensional simplicial complex, then is just the simplicial boundary map from 1-faces to 0-faces, and is the simplicial coboundary map. The matrix is symmetric, hence diagonalizable, so it has real eigenvalues (counting multiplicities). The number of nonzero eigenvalues of is , where is the number of components of .

The Matrix-Tree Theorem, first observed by Kirchhoff [22] in his work on electrical circuits (modern references include [8], [29] and [34, Chapter 5]), expresses the number of spanning trees of in terms of . The theorem has two equivalent formulations.

Theorem 1.1 (Classical Matrix-Tree Theorem).

Let be a connected graph with vertices, and let be its Laplacian matrix.

  1. If the eigenvalues of are , then

  2. For , let be the reduced Laplacian obtained from by deleting the row and column. Then

Well-known corollaries of the Matrix-Tree Theorem include Cayley’s formula [9]


where is the complete graph on vertices, and Fiedler and Sedláçek’s formula [16]


where is the complete bipartite graph on vertex sets of sizes and .

The Matrix-Tree Theorem can be refined by introducing an indeterminate for each pair of vertices , setting if do not share a common edge. The weighted Laplacian is then defined as the matrix with entries

Theorem 1.2 (Weighted Matrix-Tree Theorem).

Let be a graph with vertices, and let be its weighted Laplacian matrix.

  1. If the eigenvalues of are , then

    where is the set of all spanning trees of .

  2. For , let be the reduced weighted Laplacian obtained from by deleting the row and column. Then

By making appropriate substitutions for the indeterminates , it is often possible to obtain finer enumerative information than merely the number of spanning trees. For instance, when , introducing indeterminates and setting for all yields the Cayley-Prüfer Theorem, which enumerates spanning trees of by their degree sequences:


Note that Cayley’s formula (1) can be recovered from the Cayley-Prüfer Theorem by setting .

1.2. Simplicial spanning trees and how to count them

To extend the scope of the Matrix-Tree Theorem from graphs to simplicial complexes, we must first say what “spanning tree” means in arbitrary dimension. Kalai [20] proposed a definition that replaces the acyclicity, connectedness, and edge-count conditions with their analogues in simplicial homology. Our definition adapts Kalai’s definition to a more general class of simplicial complexes.111There are many other definitions of “simplicial tree” in the literature, depending on which properties of trees one wishes to extend; see, e.g., [4, 10, 15, 19, 28]. By adopting Kalai’s idea, we choose a definition that lends itself well to enumeration. The closest to ours in spirit is perhaps that of Masbaum and Vaintrob [28], whose main result is a Matrix-Tree-like theorem enumerating a different kind of 2-dimensional tree using Pfaffians rather than Laplacians.

Let be a -dimensional simplicial complex, and let be a subcomplex containing all faces of of dimension . We say that is a simplicial spanning tree of if the following three conditions hold:


where denotes reduced simplicial homology (for which see, e.g., [18, §2.1]). (The conditions (4a) and (4b) were introduced by Kalai in [20], while (4c) is more general, as we will explain shortly.) When , the conditions (4a)…(4c) say respectively that is acyclic, connected, and has one fewer edge than it has vertices, recovering the definition of the spanning tree of a graph. Moreover, as we will show in Proposition 3.5, any two of the three conditions together imply the third.

A graph has a spanning tree if and only if is connected. The corresponding condition for a simplicial complex of dimension  is that for all ; that is, has the rational homology type of a wedge of -dimensional spheres. We will call such a complex acyclic in positive codimension, or APC for short. This condition, which we will assume throughout the rest of the introduction, is much weaker than Cohen-Macaulayness (by Reisner’s theorem [30]), and therefore encompasses many complexes of combinatorial interest, including all connected graphs, simplicial spheres, shifted, matroid, and Ferrers complexes, and some chessboard and matching complexes.

For , let be the th simplicial boundary matrix of  (with rows and columns indexed respectively by -dimensional and -dimensional faces of ), and let be its transpose. The (th up-down) Laplacian of is ; this can be regarded either as a square matrix of size or as a linear endomorphism on ()-chains of . Define invariants

where denotes the set of all -trees of (that is, simplicial spanning trees of the -skeleton of ).

Kalai [20] studied these invariants in the case that is a simplex on vertices, and proved the formula


(of which Cayley’s formula (1) is the special case ). Kalai also proved a natural weighted analogue of (5) enumerating simplicial spanning trees by their degree sequences, thus generalizing the Cayley-Prüfer Theorem (3).

Given disjoint vertex sets (“color classes”), the faces of the corresponding complete colorful complex are those sets of vertices with no more than one vertex of each color. Equivalently, is the simplicial join of the 0-dimensional complexes . Adin [1] extended Kalai’s work by proving a combinatorial formula for , which we shall not reproduce here, for every . Note that when , the complex is a complete bipartite graph, and if for all , then is a simplex. Thus both (2) and (5) can be recovered from Adin’s formula.

Kalai’s and Adin’s beautiful formulas inspired us to look for more results about simplicial spanning tree enumeration, and in particular to formulate a simplicial version of the Matrix-Tree Theorem that could be applied to as broad a class of complexes as possible. Our first main result generalizes the Matrix-Tree Theorem to all APC simplicial complexes.

Theorem 1.3 (Simplicial Matrix-Tree Theorem).

Let be a -dimensional APC simplicial complex. Then:

  1. We have

  2. Let be the set of facets of a -SST of , and let be the reduced Laplacian obtained by deleting the rows and columns of corresponding to . Then

We will prove these formulas in Section 4.

In the special case , the number is just the number of spanning trees of the graph , recovering the classical Matrix-Tree Theorem. When , there can exist spanning trees with finite but nontrivial homology groups (the simplest example is the real projective plane). In this case, is greater than the number of spanning trees, because these “torsion trees” contribute more than 1 to the count. This phenomenon was first observed by Bolker [7], and arises also in the study of cyclotomic matroids [26] and cyclotomic polytopes [3].

The Weighted Matrix-Tree Theorem also has a simplicial analogue. Introduce an indeterminate for each facet (maximal face) , and for every set of facets define monomials and . Construct the weighted boundary matrix by multiplying each column of by , where is the facet of corresponding to that column. Let be the product of the nonzero eigenvalues of , and let

Theorem 1.4 (Weighted Simplicial Matrix-Tree Theorem).

Let be a -dimensional APC simplicial complex. Then:

  1. We have222Despite appearances, there are no missing hats on the right-hand side of this formula! Only has been replaced with its weighted analogue; is still just an integer.

  2. Let be the set of facets of a -SST of , and let be the reduced Laplacian obtained by deleting the rows and columns of corresponding to . Then

We will prove these formulas in Section 5.

Setting for all in Theorem 1.4 recovers Theorem 1.3. In fact, more is true; setting in the multiset of eigenvalues of the weighted Laplacian (reduced or unreduced) yields the eigenvalues of the corresponding unweighted Laplacian. If the complex is Laplacian integral, that is, its Laplacian matrix has integer eigenvalues, then we can hope to find a combinatorial interpretation of the factorization of furnished by Theorem 1.4. An important class of Laplacian integral simplicial complexes are the shifted complexes.

1.3. Results on shifted complexes

Let be integers, and let . A simplicial complex on vertex set is shifted if the following condition holds: whenever are vertices and is a face such that and , then . Equivalently, define the componentwise partial order on finite sets of positive integers as follows: whenever , , and for all . Then a complex is shifted precisely when it is an order ideal with respect to the componentwise partial order. (See [33, chapter 3] for general background on partially ordered sets.)

Shifted complexes were used by Björner and Kalai [5] to characterize the -vectors and Betti numbers of all simplicial complexes. Shifted complexes are also one of a small handful of classes of simplicial complexes whose Laplacian eigenvalues are known to be integral. In particular, Duval and Reiner [13, Thm. 1.1] proved that the Laplacian eigenvalues of a shifted complex on are given by the conjugate of the partition , where is the degree of vertex , that is, the number of facets containing it.

In the second part of the article, Sections 610, we study factorizations of the weighted spanning tree enumerator of under the combinatorial fine weighting

(described in more detail in Section 6), where is a -dimensional face of . Thus the term of corresponding to a particular simplicial spanning tree of contains more precise information than its vertex degrees alone (which can be recovered by further setting for all ).

For integer sets and as above, we call the ordered pair a critical pair of if , , and covers in the componentwise order. That is, for some . The long signature of is the ordered pair , where and . The corresponding -polynomial is defined as

where for each , and the operator   is defined by . (See Section 8.1 for more details, and Example 1.7 for an example.) The set of critical pairs is especially significant for a shifted family (and by extension, for a shifted complex). Since a shifted family is just an order ideal with respect to the componentwise partial order , the critical pairs identify the frontier between members and non-members of  in the Hasse diagram of . (See Example 1.7 or [23] for more details.)

Thanks to Theorem 1.4, the enumeration of SST’s of a shifted complex reduces to computing the determinant of the reduced combinatorial finely-weighted Laplacian. We show in Section 6 how this computation reduces to the computation of the eigenvalues of the algebraic finely weighted Laplacian. This modification of the combinatorial fine weighting, designed to endow the chain groups of with the structure of an algebraic chain complex, is described in detail in Section 6.3. Its eigenvalues turn out to be precisely the -polynomials associated with critical pairs.

Theorem 1.5.

Let be a -dimensional shifted complex, and let . Then the eigenvalues of the algebraic finely weighted up-down Laplacian are precisely , where ranges over all long signatures of critical pairs of -dimensional faces of .

In turn, the -polynomials are the factors of the weighted simplicial spanning tree enumerator .

Theorem 1.6.

Let be a -dimensional shifted complex with initial vertex . Then:

where ; ; and .

Theorems 1.5 and 1.6 are proved in Sections 8 and 9, respectively.

Example 1.7.

As an example to which we will return repeatedly, consider the equatorial bipyramid, the two-dimensional shifted complex with vertices and facets 123, 124, 125, 134, 135, 234, 235. A geometric realization of is shown in the figure on the left below. The figure on the right illustrates how the facets of  can be regarded as an order ideal. The boldface lines indicate critical pairs.

The Laplacian eigenvalues corresponding to the critical pairs of are as follows:

To show one of these eigenvalues in more detail,

The eigenvalues of this complex are explained in more detail in Section 8.4. Its spanning trees are enumerated in Examples 9.1 (fine weighting) and 9.3 (coarse weighting).

We prove Theorem 1.5 by exploiting the recursive structure of shifted complexes. As in [13], we begin by calculating the algebraic finely weighted eigenvalues of a near-cone in terms of the eigenvalues of its link and deletion with respect to its apex (Proposition 7.6). We can then write down a recursive formula (Theorem 8.2) for the nonzero eigenvalues of shifted complexes, thanks to their characterization as iterated near-cones, simultaneously showing that these eigenvalues must be of the form . Finally, we independently establish a recurrence (Corollary 8.8) for the long signatures of critical pairs of a shifted complex, which coincides with the recurrence for the , thus yielding a bijection between nonzero eigenvalues and critical pairs.

Corollary 8.10 shows what the eigenvalues look like in coarse weighting. Passing from weighted to unweighted eigenvalues then easily recovers the Duval-Reiner formula for Laplacian eigenvalues of shifted complexes in terms of degree sequences [13, Thm. 1.1]. Similarly, Corollary 9.2 gives the enumeration of SST’s of a shifted complex in the coarse weighting.

We are also able to show that the finely-weighted eigenvalues (though not the coarsely-weighted eigenvalues) are enough to recover the shifted complex (Corollary 8.9), or, in other words, that one can “hear the shape” of a shifted complex.

Several known results can be obtained as consequences of the general formula of Theorem 1.6.

  • The complete -skeleton of a simplex is easily seen to be shifted, and applying Theorem 1.6 to such complexes recovers Kalai’s generalization of the Cayley-Prüfer Theorem.

  • The one-dimensional shifted complexes are precisely the threshold graphs, an important class of graphs with many equivalent descriptions (see, e.g., [25]). When , Theorem 1.6 specializes to the weighted spanning tree enumerator for threshold graphs proved by Martin and Reiner [26, Thm. 4] and following from an independent result of Remmel and Williamson [31, Thm. 2.4].

  • Thanks to an idea of Richard Ehrenborg, the formula for threshold graphs can be used to recover a theorem of Ehrenborg and van Willigenburg [14], enumerating spanning trees in certain bipartite graphs called Ferrers graphs (which are not in general Laplacian integral).

We discuss these corollaries in Section 10.

Some classes of complexes that we think deserve further study include matroid complexes, matching complexes, chessboard complexes and color-shifted complexes. The first three kinds of complexes are known to be Laplacian integral, by theorems of Kook, Reiner and Stanton [24], Dong and Wachs [11], and Friedman and Hanlon [17] respectively. Every matroid complex is Cohen-Macaulay [32, §III.3], hence APC, while matching complexes and chessboard complexes are APC for certain values of their defining parameters (see [6]). Color-shifted complexes, which are a common generalization of Ferrers graphs and complete colorful complexes, are not in general Laplacian integral; nevertheless, their weighted simplicial spanning tree enumerators seem to have nice factorizations.

It is our pleasure to thank Richard Ehrenborg, Vic Reiner, and Michelle Wachs for many valuable discussions. We also thank Andrew Crites and an anonymous referee for their careful reading of the manuscript.

2. Notation and definitions

2.1. Simplicial complexes

Let be a finite set. A simplicial complex on is a family of subsets of such that

  1. ;

  2. If and , then .

The elements of are called vertices of , and the faces that are maximal under inclusion are called facets. Thus a simplicial complex is determined by its set of facets. The dimension of a face is , and the dimension of is the maximum dimension of a face (or facet). The abbreviation indicates that . We say that is pure if all facets have the same dimension; in this case, a ridge is a face of codimension 1, that is, dimension .

We write for the set of -dimensional faces of , and set . The -skeleton of is the subcomplex of all faces of dimension ,

and the pure -skeleton of is the subcomplex generated by the -dimensional faces, that is,

We assume that the reader is familiar with simplicial homology; see, e.g., [18, §2.1]. Let be a simplicial complex and . Let be a ring (if unspecified, assumed to be ), and let be the simplicial chain group of , i.e., the free -module with basis . We denote the simplicial boundary and coboundary maps respectively by

where we have identified cochains with chains via the natural inner product. We will abbreviate the subscripts in the notation for boundaries and coboundaries whenever no ambiguity can arise. We will often regard (resp. ) as a matrix whose columns and rows (resp. rows and columns) are indexed by and respectively. The (reduced) homology group of is , and the (reduced) Betti number is the rank of the largest free -module summand of .

2.2. Combinatorial Laplacians

We adopt the notation of [13] for the Laplacian operators (or, equivalently, matrices) of a simplicial complex. We summarize the notation and mention some fundamental identities here.

We will often work with multisets (of eigenvalues or of vertices), in which each element occurs with some non-negative integer multiplicity. For brevity, we drop curly braces and commas when working with multisets of integers: for instance, 5553 denotes the multiset in which 5 occurs with multiplicity three and 3 occurs with multiplicity one. The cardinality of a multiset is the sum of the multiplicities of its elements; thus . We write to mean that the multisets and differ only in their respective multiplicities of zero; for instance, . Of course, is an equivalence relation. The union operation on multisets is understood to add multiplicities: for instance, .

For , define linear operators , , on the vector space by

The spectrum of is the multiset of its eigenvalues (including zero); we define and similarly. Since each Laplacian operator is represented by a symmetric matrix, it is diagonalizable, so

The various Laplacian spectra are related by the identities

[13, eqn. (3.6)]. Therefore, each of the three families of multisets

determines the other two, and we will feel free to work with whichever one is most convenient in context.

Combinatorial Laplacians and their spectra have been investigated for a number of classes of simplicial complexes. In particular, it is known that chessboard [17], matching [11], matroid [24], and shifted [13] complexes are Laplacian integral, i.e., all their Laplacian eigenvalues are integers. Understanding which complexes are Laplacian integral is an open question. As we will see, Laplacian eigenvalues and spanning tree enumerators are inextricably linked.

3. Simplicial spanning trees

In this section, we generalize the notion of a spanning tree to arbitrary dimension using simplicial homology, following Kalai’s idea. Our definition makes sense for any ambient complex that satisfies the relatively mild APC condition.

Definition 3.1.

Let be a simplicial complex, and let . A -dimensional simplicial spanning tree (for short, SST or -SST) of is a -dimensional subcomplex such that and


We write for the set of all -SST’s of , omitting the subscript if . Note that for all .

A zero-dimensional SST is just a vertex of . If is a 1-dimensional simplicial complex on vertices—that is, a graph—then the definition of 1-SST coincides with the usual definition of a spanning tree of a graph: namely, a subgraph of which is connected, acyclic, and has edges. Next, we give a few examples in higher dimensions.

Example 3.2.

If is a simplicial sphere (for instance, the boundary of a simplicial polytope), then deleting any facet of while keeping its ()-skeleton intact produces a -SST. Therefore .

Example 3.3.

In dimension , spanning trees need not be -acyclic, merely -acyclic. For example, let be a triangulation of the real projective plane, so that , , and . Then satisfies the conditions of Definition 3.1 and is a 2-SST of itself (in fact, the only such).

Example 3.4.

Consider the equatorial bipyramid  of Example 1.7. A 2-SST of can be constructed by removing two facets , provided that contains neither of the vertices 4,5. A simple count shows that there are 15 such pairs , so .

Before proceeding any further, we show that Definition 3.1 satisfies a “two-out-of-three theorem” akin to that for spanning trees of graphs.

Proposition 3.5.

Let be a -dimensional subcomplex with . Then any two of the conditions (6a), (6b), (6c) together imply the third.


First, note that


Next, we use the standard fact that the Euler characteristic can be calculated as the alternating sum either of the -numbers or of the Betti numbers. Thus


and on the other hand,


Equating (8) and (9) gives

or equivalently

Since (6a) says that (note that must be free abelian) and (6b) says that , the conclusion follows. ∎

Definition 3.6.

A simplicial complex is acyclic in positive codimension, or APC for short, if for all .

Equivalently, a complex is APC if it has the homology type of a wedge of zero or more -dimensional spheres. In particular, any Cohen-Macaulay complex is APC. The converse is very far from true, because, for instance, an APC complex need not even be pure. For our purposes, the APC complexes are the “correct” simplicial analogues of connected graphs for the following reason.

Proposition 3.7.

For any simplicial complex , the following are equivalent:

  1. is APC.

  2. has a -dimensional spanning tree.

  3. has a -dimensional spanning tree for every .


It is trivial that (3) implies (2). To see that (2) implies (1), suppose that has a -dimensional spanning tree . Then for all , so for all . Moreover, in the diagram

we have and , so there is a surjection , implying that is APC.

To prove (1) implies (3), it suffices to consider the case , because any skeleton of an APC complex is also APC. We can construct a -SST by the following algorithm. Let . If , then there is some nonzero linear combination of facets of that is mapped to zero by . Let be one of those facets, and let . Then and for , and by the Euler characteristic formula, we have as well. Replacing with and repeating, we eventually arrive at the case , when is a -SST of . ∎

The APC condition is a fairly mild one. For instance, any -acyclic complex is clearly APC (and is its own unique SST), as is any Cohen-Macaulay complex (in particular, any shifted complex).

4. Simplicial analogues of the Matrix-Tree Theorem

We now explain how to enumerate simplicial spanning trees of a complex using its Laplacian. Throughout this section, let be an APC simplicial complex on vertex set . For , define

We are interested in the relationships between these two families of invariants. When , the relationship is given by Theorem 1.1. In the notation just defined, part (1) of that theorem says that , and part (2) says that (i.e., the determinant of the reduced Laplacian obtained from by deleting the row and column corresponding to any vertex ).

The results of this section generalize both parts of the Matrix-Tree Theorem from graphs to all APC complexes . Our arguments are closely based on those used by Kalai [20] and Adin [1] to enumerate SST’s of skeletons of simplices and of complete colorful complexes.

We begin by setting up some notation. Abbreviate , , and . Let be a set of facets of of cardinality , and let be a set of ridges such that . Define

and let be the square submatrix of with rows indexed by and columns indexed by .

Proposition 4.1.

The matrix is nonsingular if and only if and .


We may regard as the top boundary map of the -dimensional relative complex . So is nonsingular if and only if . Consider the long exact sequence


If , then and cannot both be zero. This proves the “only if” direction.

If , then (since ), so (10) implies . Therefore is a -tree, because it has the correct number of facets. Hence is finite. Then (10) implies that is finite. In fact, it is zero because the top homology group of any complex must be torsion-free. Meanwhile, has the correct number of facets to be a -SST of , proving the “if” direction. ∎

Proposition 4.2.

If is nonsingular, then


As before, we interpret as the boundary map of the relative complex . So is a map from to , and is a finite abelian group of order . On the other hand, since has no faces of dimension , its lower boundary maps are all zero, so . Since is finite, the desired result now follows from the piece


of the long exact sequence (10). ∎

We can now prove the first version of the Simplicial Matrix-Tree Theorem, relating the quantities and . Abbreviate .

Theorem 1.3 (Simplicial Matrix-Tree Theorem).

Let be an APC simplicial complex. Then:

  1. We have

  2. Let be the set of facets of a -SST of , and let denote the reduced Laplacian333A warning: This notation for reduced Laplacians specifies which rows and columns to exclude (in analogy to the notation in the statement of Theorem 1.1), in contrast to the notation for restricted boundary maps, which specifies which rows and columns to include. obtained by deleting the rows and columns of corresponding to . Then

Proof of Theorem 1.3 (1).

The Laplacian is a square matrix with rows and columns, and rank (because is APC). Let be its characteristic polynomial (where is an identity matrix), so that , the product of the nonzero eigenvalues of , is given (up to sign) by the coefficient of in . Equivalently,


where in each summand. By the Binet-Cauchy formula, we have


Combining (12) and (13), applying Proposition 4.1, and interchanging the sums, we obtain

and now applying Proposition 4.2 yields

as desired. ∎

In order to prove the “reduced Laplacian” part of Theorem 1.3, we first check that when we delete the rows of corresponding to a -SST, the resulting reduced Laplacian has the correct size, namely, that of a -SST.

Lemma 4.3.

Let be the set of facets of a