A graph $G$ is connected if and only if
the second-largest eigenvalue $\lambda_2$ of 
the [Laplacian][1] of $G$ is greater than zero.
(See, e.g.,
the <a href="http://en.wikipedia.org/wiki/Algebraic_connectivity">Wikipedia article on algebraic connectivity</a>.)

> Is there an analogous statement for
the eigenvalue $\lambda_2(M)$ of the [Laplacian operator][2]
$\Delta$ for an $n$-dimensional <strike>connected,</strike> 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!

<b>Addendum</b>. See Richard Montgomery's interesting new comment on the Laplacian on the integer lattice.


  [1]: http://en.wikipedia.org/wiki/Laplacian_matrix
  [2]: http://en.wikipedia.org/wiki/Laplace_operator