The NF-operator and the NF-Numbers of Simplicial Complexes
This work contributes to the theoretical understanding of algebraic topology and graph theory by determining specific NF-numbers for various graph families, which is of interest to researchers in pure mathematics.
This paper investigates the NF-operator and NF-numbers of simplicial complexes, providing explicit formulas for the NF-number in several families of graphs. Specifically, it computes the NF-number for dumbbell graphs, proves it is m+n+2 for complete split graphs S_n,m, and p+q+4 for double stars D_p+q.
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.