Standard monomials of $1$-skeleton ideals of multigraphs

Abstract

Given a graph $G$ on the vertex set $\{0,1,\dots ,n\}$ with the root vertex $0$, Postnikov and Shapiro associated a monomial ideal ${\mathcal{M}}_G$ in the polynomial ring $R=\mathbb{K}[x_1,\dots ,x_n]$ over a field $\mathbb{K}$ such that ${\dim }_{\mathbb{K}}(R/{\mathcal{M}}_G)=\det {\tilde{L}}_G$, where ${\tilde{L}}_G$ 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 ${\dim }_{\mathbb{K}}(R/{\mathcal{M}}_G^{(1)})\ge \det {\tilde{Q}}_G$, where ${\tilde{Q}}_G$ 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 ${\dim }_{\mathbb{K}}(R/{\mathcal{M}}_G^{(1)})=\det {\tilde{Q}}_G$. Moreover, we give examples of subgraphs $G$ of the complete multigraph $K_{n+1}^{a,b}$, in which the equality ${\dim }_{\mathbb{K}}(R/{\mathcal{M}}_G^{(1)})=\det {\tilde{Q}}_G$ holds. We also provide a conjecture on the structure of a general multigraph satisfying the above-mentioned equality.

Major revision. Abstract and introduction modified. One more section added. Comments are welcome!