Uniform estimates of nonlinear spectral gaps

2013/08/21 by Takefumi Kondo, Kondo, Takefumi, Tetsu Toyoda +1
Computer Science · Mathematics · #05C12 #51F99 #60J10 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Graph theory and applications #Markov Chains and Monte Carlo Methods #Metric Geometry (math.MG) #math.MG #msc:05C12 #msc:51F99 #msc:60J10

paper · pdf · doi:10.48550/arxiv.1308.4493

to appear in Graphs and Combinatorics

openalex publication_date 2013/08/21 · arxiv created 2015/06/13 · arxiv updated 2015/06/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

By generalizing the path method, we show that nonlinear spectral gaps of a finite connected graph are uniformly bounded from below by a positive constant which is independent of the target metric space. We apply our result to an r-ball T_d,r in the d-regular tree, and observe that the asymptotic behavior of nonlinear spectral gaps of T_d,r as r→∞ does not depend on the target metric space, which is in contrast to the case of a sequence of expanders. We also apply our result to the n-dimensional Hamming cube H_n and obtain an estimate of its nonlinear spectral gap with respect to an arbitrary metric space, which is asymptotically sharp as n→∞.

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