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

Abstract

Let $G$ be a (multi) graph on the vertex set $V=\{0,1,\dots ,n\}$ with root $0$. The $G$-parking function ideal ${\mathcal{M}}_G$ is a monomial ideal in the polynomial ring $R=\mathbb{K}[x_1,\dots ,x_n]$ over a field $\mathbb{K}$ such that ${\dim }_{\mathbb{K}}\left(\frac{R}{{\mathcal{M}}_G}\right)=\det \left({\tilde{L}}_G\right)$, where ${\tilde{L}}_G$ is the truncated Laplace matrix of $G$ and $\det \left({\tilde{L}}_G\right)$ is the determinant of ${\tilde{L}}_G$. In other words, standard monomials of the Artinian quotient $\frac{R}{{\mathcal{M}}_G}$ correspond bijectively with the spanning trees of $G$. For $0\le k\le n-1$, the $k$-skeleton ideal ${\mathcal{M}}_G^{(k)}$ of $G$ is the monomial subideal ${\mathcal{M}}_G^{(k)}=⟨m_A:\varnothing \ne A\subseteq [n]\text{ and }|A|\le k+1⟩$ of the $G$-parking function ideal ${\mathcal{M}}_G=⟨m_A:\varnothing \ne A\subseteq [n]⟩\subseteq R$. For a simple graph $G$, Dochtermann conjectured that ${\dim }_{\mathbb{K}}\left(\frac{R}{{\mathcal{M}}_G^{(1)}}\right)\ge \det \left({\tilde{Q}}_G\right)$, where ${\tilde{Q}}_G$ is the truncated signless Laplace matrix of $G$. We show that Dochtermann conjecture holds for any (simple or multi) graph $G$ on $V$.

11 pages