论文标题
$ 1 $ - 骨骼的标准单元素多编写理想
Standard monomials of $1$-skeleton ideals of multigraphs
论文作者
论文摘要
在顶点集合$ \ {0,1，\ ldots，n \} $上的图形$ g $与root dertex $ 0 $，Postnikov和Shapiro相关的单一理想$ \ MATHCAL {M} Mathcal {M} _g $在多项式环$ r = \ Mathbb { $ \ mathbb {k} $使得$ \ dim _ {\ mathbb {k}}（r/\ mathcal {m} _g）= \ det \ det \ det \ widetilde l_g $，其中$ \ widetilde l_g $是$ g $的thunced laplacian。 Dochtermann引入了$ 1 $ -Skeleton理想$ \ Mathcal {m} _g^{（1）} $ $ \ Mathcal {M} _g $ $ \ dim _ {\ mathbb {k}}（r/\ mathcal {m} _g^{（1）}）\ ge \ det \ det \ det \ widetilde q_g $，其中$ \ widetilde q_g $ q_g $是the thuncested nctuncated lucked laplacian of $ g $。在本文中，我们表征了多数$ k_ {n+1}^{a，1} $的所有子图，特别是所有简单的图形$ g $，因此$ \ dim _ {\ m athbb {k}}}}（r/\ mathcal {mathcal {m} {m}此外，我们给出了完整的多数$ k_ {n+1}^{a，b} $的子图$ g $的示例，其中等效$ \ dim _ {\ tim _ {\ mathbb {k}}}（r/\ mathcal {m}我们还提供了满足上述平等的一般多编码结构的猜想。
Given a graph $G$ on the vertex set $\{0,1,\ldots,n\}$ with the root vertex $0$, Postnikov and Shapiro associated a monomial ideal $\mathcal{M}_G$ in the polynomial ring $R=\mathbb{K}[x_1,\ldots,x_n]$ over a field $\mathbb{K}$ such that $\dim_{\mathbb{K}}(R/\mathcal{M}_G)=\det\widetilde L_G$, where $\widetilde L_G$ is the truncated Laplacian of $G$. Dochtermann introduced the $1$-skeleton ideal $\mathcal{M}_G^{(1)}$ of $\mathcal{M}_G$ which satisfies the property that $\dim_{\mathbb{K}}(R/\mathcal{M}_G^{(1)})\ge\det\widetilde Q_G$, where $\widetilde Q_G$ is the truncated signless Laplacian of $G$. In this paper we characterize all subgraphs of the multigraph $K_{n+1}^{a,1}$, in particular all simple graphs $G$, such that $\dim_{\mathbb{K}}(R/\mathcal{M}_G^{(1)})=\det\widetilde Q_G$. Moreover, we give examples of subgraphs $G$ of the complete multigraph $K_{n+1}^{a,b}$, in which the equality $\dim_{\mathbb{K}}(R/\mathcal{M}_G^{(1)})=\det\widetilde Q_G$ holds. We also provide a conjecture on the structure of a general multigraph satisfying the above-mentioned equality.
