Related papers: Minimal Free Resolutions of the $G$-parking Functi…

Given a graph $G$, the $G$-parking function ideal $M_G$ is an artinian monomial ideal in the polynomial ring $S$ with the property that a linear basis for $S/M_G$ is provided by the set of $G$-parking functions. It follows that the…

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…

We study certain groups and ideals arising from the chip-firing game on a generalisation of graphs called pargraphs. Several well-known families of toric ideals arise as toppling ideals of pargraphs. These include ideals defining rational…

Let $G$ be an (oriented) graph on the vertex set $V = \{ 0, 1,\ldots,n\}$ with root $0$. Postnikov and Shapiro associated a monomial ideal $\mathcal{M}_G$ in the polynomial ring $ R = {\mathbb{K}}[x_1,\ldots,x_n]$ over a field $\mathbb{K}$.…

Let $G$ be a (multi) graph on the vertex set $V=\{0,1,\ldots ,n\}$ with root $0$. The $G$-parking function ideal $\mathcal{M}_G$ is a monomial ideal in the polynomial ring $R=\mathbb{K}[x_1,\ldots ,x_n]$ over a field $\mathbb{K}$ such that…

Based upon a previous work of Manjunath and Sturmfels for a finite, complete, undirected graph, and a refined algorithm by Er\"ocal, Motsak, Schreyer and Steenpa{\ss} for computing syzygies, we display a free resolution of the lattice ideal…

We will explore some properties of minimal graded free resolutions of $R/I$, where $R$ is a trivariate polynomial ring over a field and $I$ is a monomial ideal. Our focus will be to consider a specific form of the resolutions when $I$ is…

In this paper we study minimal free resolutions of some classes of monomial ideals. we first give a sufficient condition to check the minimality of the resolution obtained by the mapping cone. Using it, we obtain the Betti numbers of…

An explicit combinatorial minimal free resolution of an arbitrary monomial ideal $I$ in a polynomial ring in $n$ variables over a field of characteristic $0$ is defined canonically, without any choices, using higher-dimensional…

For a finite simple graph $G$ and an integer $r \ge 1$, the $r$-connected ideal $I_r(G)$ is the squarefree monomial ideal generated by the vertex sets of connected induced subgraphs of size $r+1$, extending the classical edge ideal. We…

The Riemann-Roch theorem on a graph G is related to Alexander duality in combinatorial commutive algebra. We study the lattice ideal given by chip firing on G and the initial ideal whose standard monomials are the G-parking functions. When…

Let $I$ be a monomial ideal in two variables generated by three monomials and let $\mathcal{R}(I)$ be its Rees ideal. We describe an algorithm to compute the minimal generating set of $\mathcal{R}(I)$. Based on the data obtained by this…

The minimum rank of a graph G is the minimum rank over all real symmetric matrices whose off-diagonal sparsity pattern is the same as that of the adjacency matrix of G. In this note we present the first exact algorithm for the minimum rank…

Call a monomial ideal M "generic" if no variable appears with the same nonzero exponent in two distinct monomial generators. Using a convex polytope first studied by Scarf, we obtain a minimal free resolution of M. Any monomial ideal M can…

Given an undirected graph $G=(V,E)$, and a designated vertex $q\in V$, the notion of a $G$-parking function (with respect to $q$) was independently developed and studied by various authors, and has recently gained renewed attention. This…

Let $R = k[x_1, \dotsc , x_n]$ denote the standard graded polynomial ring over a field $k$. We study certain classes of equigenerated monomial ideals with the property that the so-called complementary ideal has no linear relations on the…

We study rational normal curves via a connection to the chip firing game. A key technique, introduced in this article, is to interpret the defining ideal of the rational normal curve as an ideal associated to a generalisation of a cycle…

In this article we prove that every toric ideal associated with a gap-free graph $G$ has a squarefree lexicographic initial ideal. Moreover, in the particular case when the complementary graph of $G$ is chordal (i.e. when the edge ideal of…

We explicitly describe cellular minimal free resolutions of certain classes of edge ideals of weighted complete bipartite graphs based on a construction of Visscher. Specifically, we show that Visscher's construction minimally resolves all…

Let $I(G)$ be the edge ideal of a simple graph $G$. We prove that $I(G)^2$ has a linear free resolution if and only if $G$ is gap-free and reg$I(G) \le 3$. Similarly, we show that $I(G)^3$ has a linear free resolution if and only if $G$ is…