Proof of Lew's conjecture on the spectral gap of simplicial complex

Xiongfeng Zhan, Xueyi Huang, Huiqiu Lin

Published 2024-07-15Version 1

Let $X$ be a simplicial complex on vertex set $V$ of size $n$. Let $X(k)$ denote the set of all $k$-dimensional simplices of $X$, and $\mathrm{deg}_X(\sigma)=|\{\eta\in X(k+1):\sigma\subseteq \eta\}|$ denote the degree of $\sigma \in X$. A missing face in $X$ is a subset $\sigma$ of $V$ such that $\sigma\notin X$ but $\tau\in X$ for any proper subset $\tau$ of $\sigma$. Let $d$ denote the maximal dimension of a missing face of $X$, and $\mu_k(X)$ denote the $k$-th spectral gap of $X$, i.e., the smallest eigenvalue of the reduced $k$-dimensional Laplacian of $X$. In [J. Combin. Theory Ser. A 169 (2020) 105127], Lew established a lower bound for $\mu_k(X)$: $$\mu_k(X)\geq (d+1)\left(\min_{\sigma\in X(k)}\mathrm{deg}_X(\sigma)+k+1\right)-dn\geq (d+1)(k+1)-dn,$$ and further conjectured that if $\mu_k(X)=(d+1)(k+1)-dn$ for some $k$, then $X\cong (\Delta_d^{(d-1)})^{*(n-k-1)}*\Delta_{(d+1)(k+1)-dn-1}$. In this paper, we confirm Lew's conjecture.