aka Laplace-Beltrami operator, $\Delta$, $\nabla^2$

The Divergence of the Gradient of a function. It is a Self-adjoint operator (like symmetric matrices.).

The Laplacian can be interpreted as the difference between the average of a function on an infinitesimal sphere around a point and the value of the function at the point itself. It is one of the most important operators in mathematical physics, used to describe phenomena as diverse as heat Diffusion, Quantum mechanics, and Wave propagation.

$\Delta \Phi(x) = 0$ is Laplace's equation

The Eigenfunctions of the Laplacian operator are the Fourier modes/harmonics. In a manifold they give the generalization of these

Laplacian on Manifolds

See the Laplacian on a Riemannian Manifold: An Introduction toAnalysis on Manifolds, and this article for applications to Geometric deep learning

Graph laplacian

Discrete version of the Laplacian, which is also defined for general Graphs. Used in Finite difference methods.

Its Eigenvectors generalize Fourier modes to graphs