  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 $V$.
