In this paper we study in detail some spectral properties of the magnetic discrete Laplacian. We identify its form-domain, characterize the absence of essential spectrum and provide the asymptotic eigenvalue distribution.

We prove upper bounds for the first eigenvalue of the Laplacian of hypersurfaces of Euclidean space involving anisotropic mean curvatures. Then, we study the equality case and its stability.

We give an estimate on the lower bound of the first non-zero eigenvalue of the Laplacian for a closed Riemannian manifold with positive Ricci curvature in terms of the in-diameter and the lower bound of the Ricci curvature.