论文标题
$ \ MATHCAL {N} \ MATHCAL {F} $ - 简单复杂的数量
The $\mathcal{N}\mathcal{F}$-Number of a Simplicial Complex
论文作者
论文摘要
令$δ$为$ [n] $的简单综合体。 $ MATHCAL {n} \ MATHCAL {F} $ - $δ$的复合物是Simplicial Complexpect $δ_ {\ Mathcal {\ MathCal {n} \ Mathcal {f}}}（Δ）$ on $ [n] $的$ facet Ideal of $δ$等于$Δ$ $δ_ {\ Mathcal {n} \ Mathcal {f}}（δ）$。此外，对于每个$ k = 2,3，\ ldots $ \ ,,，我们介绍{\ em $ k^{th} $ $ \ $ \ nathcal {n} \ Mathcal {f} $ - complect} $ groupp} $Δ^{（k）}美元$δ^{（1）} _ {\ Mathcal {n} \ Mathcal {f}}}（δ）=δ_ {\ Mathcal {n} \ Mathcal {f}}}}（δ）$。一个人可以设置$δ^{（0）} _ {\ Mathcal {n} \ Mathcal {f}}}（δ）=δ$。 $ \ MATHCAL {N} \ MATHCAL {f} $ - $δ$的数量是最小的整数$ k> 0 $，$δ^{（k）} _ {\ Mathcal {n} \ Mathcal {n} \ Mathcal {f}}}}}（δ）\ simeqδ$。在本文中，我们对$ \ MATHCAL {N} \ MATHCAL {F} $ - 有限图的数量特别感兴趣，可以将其重新升级为尺寸的简单复合体。结果表明，$ \ nathcal {n} \ Mathcal {f} $ - 有限图$ k_n \ coprod k_m $上的$ [n + m] $上的$ [n + m] $，这是完整图的不连接结合$ k_n $在$ [n] $上的$ k_n $，$ k_m $ n $ n $ n $ n $ n $ n \ geq q e g geq q e geq q e geq q geq q q e g geq 2 $ n \ geq 2 $ n \ geq q q q Q q q q q。 \ neq（2,2）$等于$ n + m + 2 $。它的推论说，$ \ MATHCAL {N} \ MATHCAL {F} $ - $ [N + M]上的完整双分图$ k_ {n，m} $的数量也等于$ n + m + m + 2 $。
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 $n + m + 2$.
