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.
Categories: math.CO