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

2024/07/15 by Zhan, Xiongfeng, Huang, Xueyi, Lin, Huiqiu · 1 citation
#05E45 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2407.10398

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 μ_k(X) is the smallest eigenvalue of the reduced k-dimensional Laplacian of X. For any k≥ -1, Lew [J. Combin. Theory Ser. A 169 (2020) 105127] established a lower bound for μ_k(X): μ_k(X)≥ (d+1)(min_{σ∈ X(k)}°_X(σ)+k+1)-dn≥ (d+1)(k+1)-dn, where °_X(σ) 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.

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• Combinatorial Laplacians and Relative Homology of Complex Pairs

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