AI 中文总结
该论文通过上拉普拉斯算子的二阶矩缺陷，定量界定了有限纯k维单纯复形到移位族的距离，并在k=1时对应到阈值图的边交换次数问题。
AI 中文摘要
设K是顶点集[n]上的有限纯k维单纯复形，k≥1，其面族为K_k。设λ₁(K)≥λ₂(K)≥…>0是其(k-1)维上拉普拉斯算子的非零特征值，在将顶点排序满足°_K(1)≥…≥°_K(n)后，令dv_T{r}(K)为至少包含在r个面中的顶点数。一个复形是“移位的”，指将一个面中的顶点替换为该面外更小的顶点后仍得到一个面。我们证明存在一个[n]上的(k+1)元子集的移位族HH，其成员数与K_k相同，满足：1/2|K_k△HH| ≤ 1/2[∑_{r≥1}(dv_T{r}(K))² - ∑_{r}λ_r(K)²]。左侧计数了为得到HH所需交换的面，因此两个序列的二阶幂和之差的一半，界定了K_k到一个移位族的距离。Gupta等人已证明，当且仅当K同构于一个移位复形时，λ(K)=dv(K)^T，该结论源于此差值等于失败的初等移位次数的两倍。本文将该恒等式转化为定量稳定性结论，并在缺陷为零时恢复该等式刻画。对于k=1，此结论界定了达到边数相同的阈值图所需的边交换次数。
英文摘要
Let $K$ be a finite pure $k$-dimensional simplicial complex, with $k\ge1$, on the vertex set $[n]$ and with facet family $K_k$. Let $λ_1(K)\geλ_2(K)\ge\cdots>0$ be the nonzero eigenvalues of its $(k-1)$-dimensional up-Laplacian, and, after ordering the vertices so that $°_K(1)\ge\cdots\ge°_K(n)$, let $\dvT{r}(K)$ be the number of vertices contained in at least $r$ facets. A complex is \emph{shifted} if replacing a vertex of a face by a smaller vertex outside the face always yields another face. We prove that there is a shifted family $\HH$ of $(k+1)$-element subsets of $[n]$, with the same number of members as $K_k$, such that \[ \tfrac12\bigl|K_k\,\triangle\,\HH\bigr| \;\le\; \tfrac12\left[\sum_{r\ge1}\bigl(\dvT{r}(K)\bigr)^{2}-\sum_{r}λ_r(K)^{2}\right]. \] The left-hand side counts the facets that have to be exchanged to reach $\HH$; thus one half of the gap between the second power sums of the two sequences bounds the distance of $K_k$ to a shifted family. The characterization $λ(K)=\dv(K)^{\mathsf T}\iff K$ is isomorphic to a shifted complex was established in \cite{Gupta} from the identity that this gap equals twice the number of failed elementary shifts. The present paper converts that identity into a quantitative stability statement and recovers the equality characterization at zero defect. For $k=1$ this bounds the number of edge exchanges needed to reach a threshold graph with the same number of edges.
