approximating various eigenvalues and eigenfunctions on riemannian manifolds

This work is concerned with the new method of approximating various kinds of eigenvalues and eigenfunctions on riemannian manifolds using the corresponding ones on the so-called proximity graphs once suitably defined. the key contribution is to show eigenvlaues and eigenfunctions converge to those of the underlying riemannian manifold. in particular, we will be concerned with steklov eigenvalues. the focus is on exploring the aspects of geometry while highlighting computational and numerical aspects for future research directions. the findings provide insights into utilizing graph-based discretization techniques to estimate properties of differential operators on manifolds with potential implications in various fields, like quantum mechanics, general relativity, statistical mechanics, and acoustics.