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

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}$_G}\right)=\det\left(\widetilde{L}_G\right)$,where$\widetilde{L}_G$isthetruncatedLaplacematrixof$G$and$\det\left(\widetilde L_G\right)$isthedeterminantof$\widetilde L_G$. In other words, standard monomials of the Artinian quotient $\frac{R}{{\mathcal{M}}_G}$ correspond bijectively with the spanning trees of $G$. For $0\le k\le n-1$, the $k$-skeleton ideal ${\mathcal{M}}_G^{(k)}$ of $G$ is the monomial subideal $$\mathcal{M}$_G^{(k)}=\left\langle m_A:\emptyset\neq A\subseteq[n]$\text{ and }$|A|\leq k+1\right\rangle$ofthe$G$-parking function ideal $$\mathcal{M}$G=\left\langle m_A : \emptyset \neq A\subseteq[n]\right\rangle\subseteq R$\mathrm{.}Forasimplegraph$G$, Dochtermann conjectured that $\dim{\mathbb K}\left(\frac{R}{$\mathcal{M}$_G^{(1)}}\right)\geq\det\left(\widetilde{Q}_G\right)$, 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 $V$.