Abstract
Based upon a previous work of Manjunath and Sturmfels for a finite, complete, undirected graph, and a refined algorithm by Erocal, Motsak, Schreyer and Steenpas for computing syzygies, we display a free resolution of the lattice ideal associated to a finite, strongly connected, weighted, directed graph. Moreover, the resolution is minimal precisely when the digraph is strongly complete.
Highlights
The aim of this paper is to present a free resolution of the lattice ideal I(L) associated to the lattice L spanned by the columns of the Laplacian matrix L of a finite, strongly connected, weighted, directed graph G, and to show that this resolution is minimal if and only if the graph G is strongly complete
The Abelian Sandpile Model (ASM) is a game played on a finite, weighted, connected, undirected graph G with n vertices, that realizes the dynamics implicit in the discrete Laplacian matrix L of the graph, this matrix being an integer matrix that is symmetric
The game’s evolution is given by a ‘toppling’ rule: each vertex containing at least as many grains as it has neighbours distributes one grain to each of them. (This process has been called ‘sand-firing’ or ‘chip-firing’.) The ASM was introduced by Bak, Tang and Wiesenfeld [3] in the context of selforganized critical phenomena in statistical physics and has been studied extensively since
Summary
The aim of this paper is to present a free resolution of the lattice ideal I(L) associated to the lattice L spanned by the columns of the Laplacian matrix L of a finite, strongly connected, weighted, directed graph G, and to show that this resolution is minimal if and only if the graph G is strongly complete. The Abelian Sandpile Model (ASM) is a game played on a finite, weighted, connected, undirected graph G with n vertices, that realizes the dynamics implicit in the discrete Laplacian matrix L of the graph, this matrix being an integer matrix that is symmetric. In the seminal paper [7], Cori, Rossin and Salvy, enumerated the vertices of the undirected graph G using a natural metric, considered the lattice L spanned over Z by the rows of the symmetric Laplacian matrix L and introduced a so-called.
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