Standard Monomials of 1-Skeleton Ideals of Graphs and Their Signless Laplace Matrices

TL;DR

This paper explores the algebraic structure of monomial ideals associated with graphs, establishing a connection between standard monomials, spanning trees, and Laplace matrices, and proves a conjecture relating ideal dimensions to the signless Laplace matrix.

Contribution

It proves Dochtermann's conjecture for both simple and multi-graphs, linking the dimension of certain monomial ideals to determinants of truncated Laplace matrices.

Findings

Standard monomials correspond bijectively with spanning trees.

Dochtermann's conjecture holds for all (multi) graphs.

The dimension of the quotient relates to determinants of Laplace matrices.

Abstract

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 $\dim_{\mathbb K}\left(\frac{R}{\mathcal{M}_G}\right)=\det\left(\widetilde{L}_G\right)$, where $\widetilde{L}_G$ is the truncated Laplace matrix of $G$ and $\det\left(\widetilde L_G\right)$ is the determinant of $\widetilde L_G$. In other words, standard monomials of the Artinian quotient $\frac{R}{\mathcal{M}_G}$ correspond bijectively with the spanning trees of $G$. For $0\leq k\leq n-1$, the $k$-skeleton ideal $\mathcal{M}_G^{(k)}$ of $G$ is the monomial subideal $\mathcal{M}_G^{(k)}=\left\langle m_A:\emptyset\neq A\subseteq[n]\text{ and }|A|\leq k+1\right\rangle$ of the $G$-parking function ideal $\mathcal{M}_G=\left\langle m_A : \emptyset \neq A\subseteq[n]\right\rangle\subseteq R$. For a simple graph $G$, Dochtermann conjectured that $\dim_{\mathbb K}\left(\frac{R}{\mathcal{M}_G^{(1)}}\right)\geq\det\left(\widetilde{Q}_G\right)$, where $\widetilde Q_G$ is the truncated signless Laplace matrix of $G$. We show that Dochtermann conjecture holds for any (simple or multi) graph $G$ on $V$.