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We study the Bergman complex B(M) of a matroid M : a poly-hedral complex which arises in algebraic geometry, but which we describe purely combinatorially. We prove that a natural subdivision of the Bergman complex of M is a geometric realization of the order complex of its lattice of flats. In addition, we show that the Bergman fan B(K n) of the graphical… (More)

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… (More)

We study the class of matroids whose independent set complexes are shifted simplicial complexes. We prove two characterization theorems, one of which is constructive. In addition, we show this class is closed under taking minors and duality. Finally, we give results on shifted broken circuit complexes.

We consider chip-firing dynamics defined by arbitrary M-matrices. M-matrices generalize graph Laplacians and were shown by Gabrielov to yield avalanche finite systems. Building on the work of Baker and Shokrieh, we extend the concept of energy minimizing chip configurations. Given an M-matrix, we show that there exists a unique energy minimizing… (More)

We consider projections of points onto fundamental chambers of finite real reflection groups. Our main result shows that for groups of type An, Bn, and Dn, the coefficients of the characteristic polynomial of the reflection arrangement are proportional to the spherical volumes of the sets of points that are projected onto faces of a given dimension. We also… (More)

We study, in three parts, degree sequences of k-families (or k-uniform hypergraphs) and shifted k-families. • The first part collects for the first time in one place, various implications such as Threshold ⇒ Uniquely Realizable ⇒ Degree-Maximal ⇒ Shifted which are equivalent concepts for 2-families (= simple graphs), but strict implications for k-families… (More)

We consider a variety of connections between threshold graphs, shifted complexes, and simplicial complexes naturally formed from a graph. These graphical complexes include the independent set, neighborhood, and dominance complexes. We present a number of structural results and relations among them including new characterizations of the class of threshold… (More)

To recognize three-dimensional objects it is important to model how their appearances can change due to changes in viewpoint. A key aspect of this involves understanding which object features can be simultaneously visible under different viewpoints. We address this problem in an image-based framework, in which we use a limited number of images of an object… (More)

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 enumera-tors and Laplacian eigenvalues of cubes; the latter are integers. We prove a… (More)

We generalize the theory of critical groups from graphs to sim-plicial complexes. Specifically, given a simplicial complex, we define a family of abelian groups in terms of combinatorial Laplacian operators, generalizing the construction of the critical group of a graph. We show how to realize these critical groups explicitly as cokernels of reduced… (More)