Let $\bigtriangleup$ be a simplicial complex and let $δ_{\mathcal{NF}}$ denote the NF-operator. The NF-complex $δ_{\mathcal{NF}}(\bigtriangleup)$ is defined as the Stanley--Reisner complex of the facet ideal of $\bigtriangleup$. Iterating $δ_{\mathcal{NF}}$ gives a periodic orbit (up to isomorphism), and the smallest positive integer $t$ for which $δ_{\mathcal{NF}}^{\,t}(\bigtriangleup)\cong \bigtriangleup$ is called the \emph{NF-number} of $\bigtriangleup$ (Habi and Mahmood, Algebra Colloquium, 2022). In this work, we provide various results and determine explicit formulas for the NF-number for several families of graphs. In particular, we compute the NF-number for dumbbell graphs. We also prove that the NF-number of the complete split graph $S_{n,m}$ equals $m+n+2$, and that the NF-number of the double star $D_{p+q}$ equals $p+q+4$. We conclude with remarks, open problems, and conjectures to guide future research.

翻译：设$\bigtriangleup$为一单纯复形，$δ_{\mathcal{NF}}$表示NF-算子。NF-复形$δ_{\mathcal{NF}}(\bigtriangleup)$定义为$\bigtriangleup$的面理想的Stanley–Reisner复形。迭代应用$δ_{\mathcal{NF}}$产生周期性轨道（在同构意义下），使$δ_{\mathcal{NF}}^{\,t}(\bigtriangleup)\cong \bigtriangleup$成立的最小正整数$t$称为$\bigtriangleup$的\textit{NF-数}（Habi和Mahmood, Algebra Colloquium, 2022）。本文给出了若干结果，并确定了多类图的NF-数的显式公式。特别地，我们计算了哑铃图的NF-数，证明了完全分裂图$S_{n,m}$的NF-数等于$m+n+2$，以及双星图$D_{p+q}$的NF-数等于$p+q+4$。最后，我们附上评注、开放问题及猜想以指导未来研究。