Simplicial matrix-tree theorems

Abstract

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 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.

RANDOM COMPLEXES AND ℓ2-BETTI NUMBERS

“…By straightforward applications of the Cauchy-Binet identity as in Duval, Klivans, and Martin [12], we obtain the following corollary: …”

“…Let Γ := (X T , X S c ). As in Proposition 4.1 of [12], we have H d (Γ) = 0 since ∂ S,T is nonsingular. As in Proposition 4.2 of [12], we also have that |det ∂ S,T | = |H d−1 (Γ)|.…”

“…Cayley's theorem was extended to higher dimensions by Kalai [22], who showed that a certain enumeration of k-dimensional complexes in an (n − 1)-dimensional simplex resulted in n to the power n−2 k . An extension of Kalai's result to general simplicial complexes was given by Duval, Klivans, and Martin [12]; an extension in a different direction was given by Adin [1]. There is more than one natural way to extend the notion of spanning tree to higher dimensions; we choose a slightly different one than the choice of Duval, Klivans, and Martin [12].…”

A higher Boltzmann distribution

“…Kalai's subcomplexes are higher dimensional analogs of spanning trees, since when k ¼ 1, they are just the spanning trees of the 1-skeleton of D n . Subsequent work in this area extended Kalai's result to more general complexes (Catanzaro et al 2015;Cappell and Miller 2015;Duval et al 2009Duval et al , 2011Duval et al , 2015Lyons 2009). With respect to these efforts, the notion of spanning tree was extended to higher dimensions (cf.…”

The Looping Rate and Sandpile Density of Planar Graphs

“…The Matrix-Tree Theorem has been further generalized in a variety of ways [7,18,12,13,19]. Of interest to us here is a formula for counting (unrooted) spanning forests.…”