Stability of Shifted Complexes via the Second-Moment Defect of the Up-Laplacian

Abstract

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 $\lambda_1(K)\ge\lambda_2(K)\ge\cdots>0$ be the nonzero eigenvalues of its $(k-1)$-dimensional up-Laplacian, and, after ordering the vertices so that $\deg_K(1)\ge\cdots\ge\deg_K(n)$, 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 \[ \tfrac12\bigl|K_k\,\triangle\,\HH\bigr| \;\le\; \tfrac12\left[\sum_{r\ge1}\bigl(\dvT{r}(K)\bigr)^{2}-\sum_{r}\lambda_r(K)^{2}\right]. \] 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 $\lambda(K)=\dv(K)^{\mathsf T}\iff K$ 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.