arXiv:2611.13623v1 Announce Type: cross Abstract: Under i.i.d. uniform sampling from a closed Riemannian manifold embedded in Euclidean space, we derive first-order asymptotic expansions for the eigenvalues and eigenvectors of symmetrically normalized graph Laplacians constructed via Gaussian kernels and provide concentration inequalities for the remainder terms. As a direct consequence, we establish the corresponding weak convergence results. Unlike most related works, our analysis does not impose a multiplicity-one assumption on the eigenvalues, thereby extending the applicability of the framework to general manifolds with potentially nontrivial eigenvalue multiplicities.