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 (Formula presented.)-ball (Formula presented.) in the (Formula presented.)-regular tree, and observe that the asymptotic behavior of nonlinear spectral gaps of (Formula presented.) as (Formula presented.) 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 (Formula presented.)-dimensional Hamming cube (Formula presented.) and obtain an estimate of its nonlinear spectral gap with respect to an arbitrary metric space, which is asymptotically sharp as (Formula presented.).
- Nonlinear spectral gap
- Path method
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics