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

TL;DR

This paper proves Lew's conjecture by identifying the unique simplicial complex that attains the lower bound of spectral gaps, advancing understanding of higher-dimensional Laplacians in simplicial complexes.

Contribution

It confirms Lew's conjecture by characterizing the unique complex achieving the spectral gap lower bound, a significant step in spectral combinatorics of simplicial complexes.

Findings

Identified the unique simplicial complex reaching the spectral gap lower bound.

Confirmed Lew's conjecture on spectral gap bounds.

Enhanced understanding of spectral properties in higher-dimensional complexes.

Abstract

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 vertex set $V$ of size $n$, and let $X(k)$ denote the set of all $k$-dimensional simplices of $X$. The $k$-th spectral gap $\mu_k(X)$ is the smallest eigenvalue of the reduced $k$-dimensional Laplacian of $X$. For any $k\geq -1$, Lew [J. Combin. Theory Ser. A 169 (2020) 105127] established a lower bound for $\mu_k(X)$: $$\mu_k(X)\geq (d+1)\left(\min_{\sigma\in X(k)}\deg_X(\sigma)+k+1\right)-dn\geq (d+1)(k+1)-dn,$$ where $\deg_X(\sigma)$ and $d$ denote the degree of $\sigma$ 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.