  We study the spectral convergence of a symmetrized Graph Laplacian matrix
induced by a Gaussian kernel evaluated on pairs of embedded data, sampled from
a manifold with boundary, a sub-manifold of $\mathbb{R}^m$. Specifically, we
deduce the convergence rates for eigenpairs of the discrete Graph-Laplacian
matrix to the eigensolutions of the Laplace-Beltrami operator that are
well-defined on manifolds with boundary, including the homogeneous Neumann and
Dirichlet boundary conditions. For the Dirichlet problem, we deduce the
convergence of the \emph{truncated Graph Laplacian}, which is recently
numerically observed in applications, and provide a detailed numerical
investigation on simple manifolds. Our method of proof relies on the min-max
argument over a compact and symmetric integral operator, leveraging the RKHS
theory for spectral convergence of integral operator and a recent pointwise
asymptotic result of a Gaussian kernel integral operator on manifolds with
boundary.
