The NF-operator and the NF-Numbers of Simplicial Complexes

Let $△$ be a simplicial complex and let ${\delta }_{\mathcal{N}\mathcal{F}}$ denote the NF-operator. The NF-complex ${\delta }_{\mathcal{N}\mathcal{F}}(△)$ is defined as the Stanley–Reisner complex of the facet ideal of $△$. Iterating ${\delta }_{\mathcal{N}\mathcal{F}}$ gives a periodic orbit (up to isomorphism), and the smallest positive integer $t$ for which ${\delta }_{\mathcal{N}\mathcal{F}}^t(△)\cong △$ is called the \emph{NF-number} of $△$ (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.