55 research outputs found
Bigraphical Arrangements
We define the bigraphical arrangement of a graph and show that the
Pak-Stanley labels of its regions are the parking functions of a closely
related graph, thus proving conjectures of Duval, Klivans, and Martin and of
Hopkins and Perkinson. A consequence is a new proof of a bijection between
labeled graphs and regions of the Shi arrangement first given by Stanley. We
also give bounds on the number of regions of a bigraphical arrangement.Comment: Added Remark 19 addressing arbitrary G-parking functions; minor
revision
Primer for the algebraic geometry of sandpiles
The Abelian Sandpile Model (ASM) is a game played on a graph realizing the
dynamics implicit in the discrete Laplacian matrix of the graph. The purpose of
this primer is to apply the theory of lattice ideals from algebraic geometry to
the Laplacian matrix, drawing out connections with the ASM. An extended summary
of the ASM and of the required algebraic geometry is provided. New results
include a characterization of graphs whose Laplacian lattice ideals are
complete intersection ideals; a new construction of arithmetically Gorenstein
ideals; a generalization to directed multigraphs of a duality theorem between
elements of the sandpile group of a graph and the graph's superstable
configurations (parking functions); and a characterization of the top Betti
number of the minimal free resolution of the Laplacian lattice ideal as the
number of elements of the sandpile group of least degree. A characterization of
all the Betti numbers is conjectured.Comment: 45 pages, 14 figures. v2: corrected typo
Simplicial Dollar Game
The dollar game is a chip-firing game introduced by Baker and Norine (2007)
as a context in which to formulate and prove the Riemann-Roch theorem for
graphs. A divisor on a graph is a formal integer sum of vertices. Each
determines a dollar game, the goal of which is to transform the given divisor
into one that is effective (nonnegative) using chip-firing moves. We use Duval,
Klivans, and Martin's theory of chip-firing on simplicial complexes to
generalize the dollar game and results related to the Riemann-Roch theorem for
graphs to higher dimensions. In particular, we extend the notion of the degree
of a divisor on a graph to a (multi)degree of a chain on a simplicial complex
and use it to establish two main results. The first of these is Theorem 18,
generalizing the fact that if a divisor on a graph has large enough degree (at
least as large as the genus of the graph), it is winnable; and the second is
Corollary 34, generalizing the fact that trees (graphs of genus 0) are exactly
the graphs on which every divisor of degree 0, interpreted as an instance of
the dollar game, is winnable.Comment: 22 pages; version 2 fixes a typo in Theorem 1
Sandpiles and Dominos
We consider the subgroup of the abelian sandpile group of the grid graph
consisting of configurations of sand that are symmetric with respect to central
vertical and horizontal axes. We show that the size of this group is (i) the
number of domino tilings of a corresponding weighted rectangular checkerboard;
(ii) a product of special values of Chebyshev polynomials; and (iii) a
double-product whose factors are sums of squares of values of trigonometric
functions. We provide a new derivation of the formula due to Kasteleyn and to
Temperley and Fisher for counting the number of domino tilings of a 2m x 2n
rectangular checkerboard and a new way of counting the number of domino tilings
of a 2m x 2n checkerboard on a M\"obius strip.Comment: 35 pages, 24 figure
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