文献类型：期刊文献

英文题名：Proof of Lew's conjecture on the spectral gaps of simplicial complexes

作者：Zhan, Xiongfeng[1];Huang, Xueyi[1];Lin, Huiqiu[1]

机构：[1]East China Univ Sci & Technol, Sch Math, Shanghai 200237, Peoples R China

年份：2026

卷号：217

外文期刊名：JOURNAL OF COMBINATORIAL THEORY SERIES A

收录：;WOS:【SCI-EXPANDED(收录号:WOS:001514409300001)】；

基金：The authors would like to thank Dr. Lu Lu for valuable discussions and the anonymous referees for their insightful comments. X. Huang was supported by National Natural Science Foundation of China (No. 12471324) and Natural Science Foundation of Shanghai (No. 24ZR1415500). H. Lin was supported by National Natural Science Foundation of China (Nos. 12271162 and 12326372), Natural Science Foundation of Shanghai (Nos. 22ZR1416300 and 23JC1401500), and The Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning (No. TP2022031).

语种：英文

外文关键词：Simplicial complex; Spectral gap; Reduced k-dimensional Laplacian; Missing face; Geometric realization

摘要：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(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 >= -1, Lew (2020) [24] established a lower bound for mu(k)(X): mu(k)(X)>=(d+1)(min(sigma is an element of X(k))deg(X)(sigma)+k+1)-dn >=(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.(c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.