Standard Monomials of 1-Skeleton Ideals of Graphs and Their Signless Laplace Matrices
Chanchal Kumar, Gargi Lather, Amit Roy
TL;DR
The paper investigates G-parking function ideals $\mathcal{M}_G$ and their $k$-skeletons $\mathcal{M}_G^{(k)}$, establishing deep connections between standard monomials, parking functions, and spanning trees via truncated Laplacian matrices. It proves that for any graph $G$ on vertex set $V$, the dimension of $R/\mathcal{M}_G^{(1)}$ satisfies $\dim_{\mathbb{K}}(R/\mathcal{M}_G^{(1)}) \ge \det\widetilde{Q}_G$, and in several complete multigraph families matches the determinant exactly, by constructing and analyzing monomial ideals $\mathcal{J}_H$ for PSD matrices $H$ and applying determinant-inequality tools. A general framework based on positive semidefinite matrices $H$ yields $\dim_{\mathbb{K}}(R_n/\mathcal{J}_H) \ge \det H$, encompassing the graph-theoretic cases and tying Steck determinant formulas for various parking-function variants. Collectively, the results validate Dochtermann’s conjecture for all (simple or multi) graphs and illuminate a broad algebraic-combinatorial landscape linking monomial ideals, parking functions, and spectral graph theory.
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 $V$.