  We show both adaptive and non-adaptive minimax rates of convergence for a
family of weighted Laplacian-Eigenmap based nonparametric regression methods,
when the true regression function belongs to a Sobolev space and the sampling
density is bounded from above and below. The adaptation methodology is based on
extensions of Lepski's method and is over both the smoothness parameter
($s\in\mathbb{N}_{+}$) and the norm parameter ($M>0$) determining the
constraints on the Sobolev space. Our results extend the non-adaptive result in
\cite{green2021minimax}, established for a specific normalized graph Laplacian,
to a wide class of weighted Laplacian matrices used in practice, including the
unnormalized Laplacian and random walk Laplacian.
