  The manifold Helmholtzian (1-Laplacian) operator $\Delta_1$ elegantly
generalizes the Laplace-Beltrami operator to vector fields on a manifold
$\mathcal M$. In this work, we propose the estimation of the manifold
Helmholtzian from point cloud data by a weighted 1-Laplacian $\mathbf{\mathcal
L}_1$. While higher order Laplacians ave been introduced and studied, this work
is the first to present a graph Helmholtzian constructed from a simplicial
complex as an estimator for the continuous operator in a non-parametric
setting. Equipped with the geometric and topological information about
$\mathcal M$, the Helmholtzian is a useful tool for the analysis of flows and
vector fields on $\mathcal M$ via the Helmholtz-Hodge theorem. In addition, the
$\mathbf{\mathcal L}_1$ allows the smoothing, prediction, and feature
extraction of the flows. We demonstrate these possibilities on substantial sets
of synthetic and real point cloud datasets with non-trivial topological
structures; and provide theoretical results on the limit of $\mathbf{\mathcal
L}_1$ to $\Delta_1$.
