Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Standard Monomials of 1-Skeleton Ideals of Graphs and Their Signless
  Laplace Matrices   Let $G$ be a (multi) graph on the vertex set $V=\{0,1,\ldots ,n\}$ 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 $V$.

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The categories of this paper are: eess
7
The categories of this paper are: hep-th
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The categories of this paper are: patt-sol
5
The categories of this paper are: gr-qc
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The categories of this paper are: comp-gas
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The categories of this paper are: dg-ga
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The categories of this paper are: acc-phys
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The categories of this paper are: cond-mat
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The categories of this paper are: nlin
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The categories of this paper are: nucl-ex
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The categories of this paper are: cs
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The categories of this paper are: quant-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-lat
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The categories of this paper are: physics