  Consider $n$ points independently sampled from a density $p$ of class
$\mathcal{C}^2$ on a smooth compact $d$-dimensional sub-manifold $\mathcal{M}$
of $\mathbb{R}^m$, and consider the generator of a random walk visiting these
points according to a transition kernel $K$. We study the almost sure uniform
convergence of this operator to the diffusive Laplace-Beltrami operator when
$n$ tends to infinity. This work extends known results of the past 15 years. In
particular, our result does not require the kernel $K$ to be continuous, which
covers the cases of walks exploring $k$NN-random and geometric graphs, and
convergence rates are given. The distance between the random walk generator and
the limiting operator is separated into several terms: a statistical term,
related to the law of large numbers, is treated with concentration tools and an
approximation term that we control with tools from differential geometry. The
convergence of $k$NN Laplacians is detailed.
