A graph $G$ is connected if and only if the second-largest eigenvalue $\lambda_2$ of the Laplacian of $G$ is greater than zero. (See, e.g., the Wikipedia article on algebraic connectivity.)

Is there an analogous statement for the eigenvalue $\lambda_2(M)$ of the Laplacian operator $\Delta$ for an $n$-dimensional

~~connected,~~closed Riemannian manifold $M$?

($\Delta(f) = \nabla^2(f) = −\mathrm{div}(\mathrm{grad}(f))$.)

I am trying to understand the relationship between Laplacians on graphs and Laplacians on Riemannian manifolds. Pointers to help elucidate the connection would be greatly appreciated!

**Addendum**. See Richard Montgomery's interesting new comment on the Laplacian on the integer lattice.