Laplacian ideals, arrangements, and resolutions

TL;DR

This paper explores the algebraic structure of Laplacian ideals related to graphs, linking combinatorial graph properties with algebraic resolutions and confirming conjectures about their Betti numbers.

Contribution

It introduces a cellular resolution for the monomial ideal associated with the Laplacian lattice of a graph, generalizing previous work and connecting algebraic and combinatorial graph theories.

Findings

Resolutions supported on bounded subcomplexes of graphical arrangements

Verification of a conjecture on Betti numbers of M_G

Characterization of Betti numbers via acyclic orientations

Abstract

The Laplacian matrix of a graph G describes the combinatorial dynamics of the Abelian Sandpile Model and the more general Riemann-Roch theory of G. The lattice ideal associated to the lattice generated by the columns of the Laplacian provides an algebraic perspective on this recently (re)emerging field. This ideal I_G has a distinguished monomial initial ideal M_G, characterized by the property that the standard monomials are in bijection with the G-parking functions of the graph G. The ideal M_G was also introduced by Postnikov and Shapiro (2004) in the context of monotone monomial ideals. We study resolutions of M_G and show that a minimal free cellular resolution is supported on the bounded subcomplex of a section of the graphical arrangement of G. This generalizes constructions from Postnikov and Shapiro (for the case of the complete graph) and connects to work of Manjunath and Sturmfels, and of Perkinson et al. on the commutative algebra of Sandpiles. As a corollary we verify a conjecture of Perkinson et al. regarding the Betti numbers of M_G, and in the process provide a combinatorial characterization in terms of acyclic orientations.