A Weighted Cellular Matrix-tree Theorem, with Applications to Complete Colorful and Cubical Complexes
نویسندگان
چکیده
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.
منابع مشابه
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, pr...
متن کاملFactorizations of some weighted spanning tree enumerators
We give factorizations for weighted spanning tree enumerators of Cartesian products of complete graphs, keeping track of fine weights related to degree sequences and edge directions. Our methods combine Kirchhoff’s Matrix-Tree Theorem with the technique of identification of factors.
متن کاملSimplicial and Cubical Complexes: Analogies and Differences
The research summarized in this thesis consists essentially of two parts. In the first, we generalize a coloring theorem of Baxter about triangulations of the plane (originally used to prove combinatorially Brouwer's fixed point theorem in two dimensions) to arbitrary dimensions and to oriented simplicial and cubical pseudomanifolds. We show that in a certain sense no other generalizations may ...
متن کاملSimplicial Matrix-Tree Theorems
Abstract. 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 Lapl...
متن کاملThe Tits Alternative for Cat(0) Cubical Complexes
We prove a Tits alternative theorem for groups acting on CAT(0) cubical complexes. Namely, suppose that G is a group for which there is a bound on the orders of its finite subgroups. We prove that if G acts properly on a finitedimensional CAT(0) cubical complex, then either G contains a free subgroup of rank 2 or G is finitely generated and virtually abelian. In particular the above conclusion ...
متن کامل