Standard monomials of $1$-skeleton ideal of a graph

Abstract: Let $G$ be a (multi) graph on the vertex set $V=\{0,1,\ldots ,n\}$ with root $0$. The $G$-parking function ideal $\M_G$ is a monomial ideal in the polynomial ring $R=\mathbb{K}[x_1,\ldots ,x_n]$ over a field $\mathbb{K}$ such that dim$_{\mathbb{K}}\big(\frac{R}{\mathcal{M}_G}\big)$ $=\det\left(\widetilde{L}_G\right)$, where $\widetilde{L}_G$ is the truncated Laplace matrix of $G$. In other words, standard monomials of the Artinian quotient $\frac{R}{M_G}$ 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 a 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 $\mathcal{M}_G=\left\langle m_A:\emptyset\neq A\subseteq[n]\right\rangle\subseteq R$. In this talk we will focus on the $1$-skeleton ideal $\mathcal{M}_G^{(1)}$ of a graph $G$ and see how the number of standard monomials of $\frac{R}{\mathcal{M}_G^{(1)}}$ is related to the truncated signless Laplace matrix $\Q_G$ of $G$. This is based on joint work with Chanchal Kumar and Gargi Lather.

commutative algebraalgebraic topologycombinatorics

Audience: researchers in the topic

Applications of Combinatorics in Algebra, Topology and Graph Theory

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