I tiakina i:
| Kaituhi matua: | |
|---|---|
| Hōputu: | Preprint |
| I whakaputaina: |
2020
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| Ngā marau: | |
| Urunga tuihono: | https://arxiv.org/abs/2010.14474 |
| Ngā Tūtohu: |
Tāpirihia he Tūtohu
Kāore He Tūtohu, Me noho koe te mea tuatahi ki te tūtohu i tēnei pūkete!
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Rārangi ihirangi:
- 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.