论文标题
图形的1骨骨骼理想及其无标志的拉普拉斯矩阵的标准单元素
Standard Monomials of 1-Skeleton Ideals of Graphs and Their Signless Laplace Matrices
论文作者
论文摘要
让$ g $为顶点集合的（多）图$ v = \ {0,1，\ ldots，n \} $，带有root $ 0 $。 $ g $ -parking函数理想$ \ mathcal {m} _g $是多项式环$ r = \ mathbb {k} [x_1，\ ldots，x_n] $在字段$ \ mathbb {k} $上$ \ dim \ dim _ mathbb { k} \ left（\ frac {r} {\ mathcal {m} _g} \ right）= \ det \ left（\ widetilde {l} _g \ right）$，其中$ \ widetilde {l} _g $是截断的laplace matrix of $ g $ g $。 l_g \ right）$是$ \ widetilde l_g $的决定因素。换句话说，Artinian商的标准单元$ \ frac {r} {\ Mathcal {m} _g} $与$ g $的生成树相对应。对于$ 0 \ leq k \ leq n-1 $，$ k $ - skeleton理想$ \ mathcal {m} _g^{（k）} $ of $ g $是单一的subideal $ \ nathcal {m} _g {m} _g^{（k）} = \ lest \ left \ lest \ langle \ langle \ langle \ nemet \ nelet \ nelet A \ subseteq [n] \ text {and} | a | \ leq k+1 \ range \ rangle $的$ g $ -parking函数理想$ \ mathcal {m} _g = \ langle m_a：\ langle m_a：\ emertyset \ emertyset \ neq \ neq \ neq a \ subseteq [n n]对于简单的图$ g $，dochtermann猜想是$ \ dim _ {\ mathbb k} \ left（\ frac {r} {\ mathcal {m}是$ g $的无截断的无标志性拉普拉斯矩阵。我们表明，Dochtermann的猜想适用于$ V $上任何（简单或多）图$ G $。
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 $V$.
