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The $\mathcal{N}\mathcal{F}$-Number of a Simplicial Complex

Let $Δ$ be a simplicial complex on $[n]$. The $\mathcal{N}\mathcal{F}$-complex of $Δ$ is the simplicial complex $δ_{\mathcal{N}\mathcal{F}}(Δ)$ on $[n]$ for which the facet ideal of $Δ$ is equal to the Stanley--Reisner ideal of $δ_{\mathcal{N}\mathcal{F}}(Δ)$. Furthermore, for each $k = 2,3,\ldots$\,, we introduce {\em $k^{th}$ $\mathcal{N}\mathcal{F}$-complex} $δ^{(k)}_{\mathcal{N}\mathcal{F}}(Δ)$ which is inductively defined by $δ^{(k)}_{\mathcal{N}\mathcal{F}}(Δ) = δ_{\mathcal{N}\mathcal{F}}(δ^{(k-1)}_{\mathcal{N}\mathcal{F}}(Δ))$ with setting $δ^{(1)}_{\mathcal{N}\mathcal{F}}(Δ) = δ_{\mathcal{N}\mathcal{F}}(Δ)$. One can set $δ^{(0)}_{\mathcal{N}\mathcal{F}}(Δ) = Δ$. The $\mathcal{N}\mathcal{F}$-number of $Δ$ is the smallest integer $k > 0$ for which $δ^{(k)}_{\mathcal{N}\mathcal{F}}(Δ) \simeq Δ$. In the present paper we are especially interested in the $\mathcal{N}\mathcal{F}$-number of a finite graph, which can be regraded as a simplicial complex of dimension one. It is shown that the $\mathcal{N}\mathcal{F}$-number of the finite graph $K_n\coprod K_m$ on $[n + m]$, which is the disjoint union of the complete graphs $K_n$ on $[n]$ and $K_m$ on $[m]$, where $n \geq 2$ and $m \geq 2$ with $(n,m) \neq (2,2)$, is equal to $n + m + 2$. Its corollary says that the $\mathcal{N}\mathcal{F}$-number of the complete bipartite graph $K_{n,m}$ on $[n+m]$ is also equal to $n + m + 2$.