Proof of Lew's conjecture on the spectral gaps of simplicial complexes

As a generalization of graph Laplacians to higher dimensions, the combinatorial Laplacians of simplicial complexes have garnered increasing attention. Let X be a simplicial complex on n vertices, and let $X\left(k\right)$ denote the set of all k-dimensional simplices of X. The k-th spectral gap ${\mu }_k\left(X\right)$ is the smallest eigenvalue of the reduced k-dimensional Laplacian of X. For any $k\ge -1$, Lew (2020) [24] established a lower bound for ${\mu }_k\left(X\right)$:${\mu }_k\left(X\right)\ge (d+1)(\underset{\sigma \in X\left(k\right)}{\min }{\mathrm{deg}}_X(\sigma )+k+1)-dn\ge (d+1)(k+1)-dn,$ where ${\mathrm{deg}}_X\left(\sigma \right)$ and d denote the degree of σ in X and the maximal dimension of a missing face of X, respectively. In this paper, we identify the unique simplicial complex that achieves the lower bound of the k-th spectral gap, $(d+1)(k+1)-dn$, for some k, thereby confirming a conjecture proposed by Lew.