3,408 research outputs found
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.
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 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
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
Melonic phase transition in group field theory
Group field theories have recently been shown to admit a 1/N expansion
dominated by so-called `melonic graphs', dual to triangulated spheres. In this
note, we deepen the analysis of this melonic sector. We obtain a combinatorial
formula for the melonic amplitudes in terms of a graph polynomial related to a
higher dimensional generalization of the Kirchhoff tree-matrix theorem. Simple
bounds on these amplitudes show the existence of a phase transition driven by
melonic interaction processes. We restrict our study to the Boulatov-Ooguri
models, which describe topological BF theories and are the basis for the
construction of four dimensional models of quantum gravity.Comment: 8 pages, 4 figures; to appear in Letters in Mathematical Physic
Weighted Tree-Numbers of Matroid Complexes
International audienceWe give a new formula for the weighted high-dimensional tree-numbers of matroid complexes. This formula is derived from our result that the spectra of the weighted combinatorial Laplacians of matroid complexes consist of polynomials in the weights. In the formula, Crapo’s -invariant appears as the key factor relating weighted combinatorial Laplacians and weighted tree-numbers for matroid complexes.Nous présentons une nouvelle formule pour les nombres d’arbres pondérés de grande dimension des matroïdes complexes. Cette formule est dérivée du résultat que le spectre des Laplaciens combinatoires pondérés des matrides complexes sont des polynômes à plusieurs variables. Dans la formule, le ;-invariant de Crapo apparaît comme étant le facteur clé reliant les Laplaciens combinatoires pondérés et les nombres d’arbres pondérés des matroïdes complexes
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
Cyclotomic and simplicial matroids
Two naturally occurring matroids representable over Q are shown to be dual:
the {\it cyclotomic matroid} represented by the roots of unity
inside the cyclotomic extension ,
and a direct sum of copies of a certain simplicial matroid, considered
originally by Bolker in the context of transportation polytopes. A result of
Adin leads to an upper bound for the number of -bases for among
the roots of unity, which is tight if and only if has at most two
odd prime factors. In addition, we study the Tutte polynomial of in the
case that has two prime factors.Comment: 9 pages, 1 figur
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