274 research outputs found
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
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
Cuts and flows of cell complexes
We study the vector spaces and integer lattices of cuts and flows associated
with an arbitrary finite CW complex, and their relationships to group
invariants including the critical group of a complex. Our results extend to
higher dimension the theory of cuts and flows in graphs, most notably the work
of Bacher, de la Harpe and Nagnibeda. We construct explicit bases for the cut
and flow spaces, interpret their coefficients topologically, and give
sufficient conditions for them to be integral bases of the cut and flow
lattices. Second, we determine the precise relationships between the
discriminant groups of the cut and flow lattices and the higher critical and
cocritical groups with error terms corresponding to torsion (co)homology. As an
application, we generalize a result of Kotani and Sunada to give bounds for the
complexity, girth, and connectivity of a complex in terms of Hermite's
constant.Comment: 30 pages. Final version, to appear in Journal of Algebraic
Combinatoric
Fundamental polytopes of metric trees via parallel connections of matroids
We tackle the problem of a combinatorial classification of finite metric
spaces via their fundamental polytopes, as suggested by Vershik in 2010. In
this paper we consider a hyperplane arrangement associated to every split
pseudometric and, for tree-like metrics, we study the combinatorics of its
underlying matroid. We give explicit formulas for the face numbers of
fundamental polytopes and Lipschitz polytopes of all tree-like metrics, and we
characterize the metric trees for which the fundamental polytope is simplicial.Comment: 20 pages, 2 Figures, 1 Table. Exposition improved, references and new
results (last section) adde
Decision trees, monotone functions, and semimatroids
We define decision trees for monotone functions on a simplicial complex. We
define homology decidability of monotone functions, and show that various
monotone functions related to semimatroids are homology decidable. Homology
decidability is a generalization of semi-nonevasiveness, a notion due to
Jonsson. The motivating example is the complex of bipartite graphs, whose Betti
numbers are unknown in general.
We show that these monotone functions have optimum decision trees, from which
we can compute relative Betti numbers of related pairs of simplicial complexes.
Moreover, these relative Betti numbers are coefficients of evaluations of the
Tutte polynomial, and every semimatroid collapses onto its broken circuit
complex.Comment: 16 page
Affine Buildings and Tropical Convexity
The notion of convexity in tropical geometry is closely related to notions of
convexity in the theory of affine buildings. We explore this relationship from
a combinatorial and computational perspective. Our results include a convex
hull algorithm for the Bruhat--Tits building of SL and techniques for
computing with apartments and membranes. While the original inspiration was the
work of Dress and Terhalle in phylogenetics, and of Faltings, Kapranov, Keel
and Tevelev in algebraic geometry, our tropical algorithms will also be
applicable to problems in other fields of mathematics.Comment: 22 pages, 4 figure
Generic Cohen-Macaulay monomial ideals
Given a simplicial complex, it is easy to construct a generic deformation of
its Stanley-Reisner ideal. The main question under investigation in this paper
is how to characterize the simplicial complexes such that their Stanley-Reisner
ideals have Cohen-Macaulay generic deformations. Algorithms are presented to
construct such deformations for matroid complexes, shifted complexes, and tree
complexes.Comment: 18 pages, 8 figure
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.
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