  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 $r\to\infty$ 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 $n\to\infty$.
