The largest Laplacian eigenvalue and the balancedness of simplicial complexes

2024/05/29 by Yi-Zheng Fan, Fan, Yi-Zheng, Hui-Feng Wu +3 · 3 citations
Computer Science · Mathematics · #05C65 #05E45 #47J10 #55U05 #Combinatorics (math.CO) #FOS: Mathematics #Geometric and Algebraic Topology #Graph theory and applications #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2405.19078

openalex publication_date 2024/05/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K be a simplical complex, and let L_i^up(K), Q_i^up(K) be the i-th up Laplacian and signless Laplacian of K, respectively. In this paper we proved that the largest eigenvalue of L_i^up(K) is not greater than the largest eigenvalue of Q_i^up(K); furthermore, if K is (i+1)-path connected, then the equality holds if and only if the i-th incidence signed graph B_i(K) of K is balanced. As an application we provided an upper bound for the largest eigenvalue of the i-th up Laplacian of K, which improves the bound given by Horak and Jost and generalizes the result of Anderson and Morley on graphs.We characterized the balancedness of simplicial complexes under operations such as wedge sum, join, Cartesian product and duplication of motifs. For each i ≥ 0, by using wedge sum or duplication of motifs, we can construct an infinitely many (i+1)-path connected simplicial complexes K with B_i(K) being balanced.

Cited by

• The Smith normal form of Laplacian matrices of simplicial annuli and high dimensional trees
• Sums of Laplacian eigenvalues and sums of degrees
• Spectral radius of simplicial complexes without holes

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