33 research outputs found
Cellular spanning trees and Laplacians of cubical complexes
We prove a Matrix-Tree Theorem enumerating the spanning trees of a cell
complex in terms of the eigenvalues of its cellular Laplacian operators,
generalizing a previous result for simplicial complexes. As an application, we
obtain explicit formulas for spanning tree enumerators and Laplacian
eigenvalues of cubes; the latter are integers. We prove a weighted version of
the eigenvalue formula, providing evidence for a conjecture on weighted
enumeration of cubical spanning trees. We introduce a cubical analogue of
shiftedness, and obtain a recursive formula for the Laplacian eigenvalues of
shifted cubical complexes, in particular, these eigenvalues are also integers.
Finally, we recover Adin's enumeration of spanning trees of a complete colorful
simplicial complex from the cellular Matrix-Tree Theorem together with a result
of Kook, Reiner and Stanton.Comment: 24 pages, revised version, to appear in Advances in Applied
Mathematic
Simplicial matrix-tree theorems
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.Comment: 36 pages, 2 figures. Final version, to appear in Trans. Amer. Math.
So
Simplicial and Cellular Trees
Much information about a graph can be obtained by studying its spanning
trees. On the other hand, a graph can be regarded as a 1-dimensional cell
complex, raising the question of developing a theory of trees in higher
dimension. As observed first by Bolker, Kalai and Adin, and more recently by
numerous authors, the fundamental topological properties of a tree --- namely
acyclicity and connectedness --- can be generalized to arbitrary dimension as
the vanishing of certain cellular homology groups. This point of view is
consistent with the matroid-theoretic approach to graphs, and yields
higher-dimensional analogues of classical enumerative results including
Cayley's formula and the matrix-tree theorem. A subtlety of the
higher-dimensional case is that enumeration must account for the possibility of
torsion homology in trees, which is always trivial for graphs. Cellular trees
are the starting point for further high-dimensional extensions of concepts from
algebraic graph theory including the critical group, cut and flow spaces, and
discrete dynamical systems such as the abelian sandpile model.Comment: 39 pages (including 5-page bibliography); 5 figures. Chapter for
forthcoming IMA volume "Recent Trends in Combinatorics
A non-partitionable Cohen-Macaulay simplicial complex
A long-standing conjecture of Stanley states that every Cohen-Macaulay
simplicial complex is partitionable. We disprove the conjecture by constructing
an explicit counterexample. Due to a result of Herzog, Jahan and Yassemi, our
construction also disproves the conjecture that the Stanley depth of a monomial
ideal is always at least its depth.Comment: Final version. 13 pages, 2 figure
A weighted cellular matrix-tree theorem, with applications to complete colorful and cubical complexes
We present a version of the weighted cellular matrix-tree theorem that is
suitable for calculating explicit generating functions for spanning trees of
highly structured families of simplicial and cell complexes. We apply the
result to give weighted generalizations of the tree enumeration formulas of
Adin for complete colorful complexes, and of Duval, Klivans and Martin for
skeleta of hypercubes. We investigate the latter further via a logarithmic
generating function for weighted tree enumeration, and derive another
tree-counting formula using the unsigned Euler characteristics of skeleta of a
hypercube and the Crapo -invariant of uniform matroids.Comment: 22 pages, 2 figures. Sections 6 and 7 of previous version simplified
and condensed. Final version to appear in J. Combin. Theory Ser.
The Partitionability Conjecture
This is the authors' accepted manuscript. First published in Notices of the American Mathematical Society Volume 64 Issue 2, 2017, published by the American Mathematical Society.In 1979 Richard Stanley made the following conjecture: Every Cohen-Macaulay simplicial complex is partitionable. Motivated by questions in the theory of face numbers of simplicial complexes, the Partitionability Conjecture sought to connect a purely combinatorial condition (partitionability) with an algebraic condition (Cohen-Macaulayness). The algebraic combinatorics community widely believed the conjecture to be true, especially in light of related stronger conjectures and weaker partial results. Nevertheless, in a 2016 paper [DGKM16], the three of us (Art, Carly, and Jeremy), together with Jeremy's graduate student Bennet Goeckner, constructed an explicit counterexample. Here we tell the story of the significance and motivation behind the Partitionability Conjecture and its resolution. The key mathematical ingredients include relative simplicial complexes, nonshellable balls, and a surprise appearance by the pigeonhole principle. More broadly, the narrative theme of modern algebraic combinatorics: to understand discrete structures through algebraic, geometric, and topological lenses
Simplicial effective resistance and enumeration of spanning trees
A graph can be regarded as an electrical network in which each edge is a
resistor. This point of view relates combinatorial quantities, such as the
number of spanning trees, to electrical ones such as effective resistance. The
second and third authors have extended the combinatorics/electricity analogy to
higher dimension and expressed the simplicial analogue of effective resistance
as a ratio of weighted tree enumerators. In this paper, we first use that ratio
to prove a new enumeration formula for color-shifted complexes, confirming a
conjecture by Aalipour and the first author, and generalizing a result of
Ehrenborg and van Willigenburg on Ferrers graphs. We then use the same
technique to recover an enumeration formula for shifted complexes, first proved
by Klivans and the first and fourth authors. In each case, we add facets one at
a time, and give explicit expressions for simplicial effective resistances of
added facets by constructing high-dimensional analogues of currents and
voltages (respectively homological cycles and cohomological cocycles).Comment: 27 pages, minor revisions from v