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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.

2 Removed "connected."

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 $M$?

($\Delta(f) = \nabla^2(f) = −\mathrm{div}(\mathrm{grad}(f))$.)Of course, $M$ is already connected, so the analog, if it exists, cannot be that naively straightforward.

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!

1

# Laplacians on graphs vs. Laplacians on Riemannian manifolds: $\lambda_2$?

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 $M$?

($\Delta(f) = \nabla^2(f) = −\mathrm{div}(\mathrm{grad}(f))$.) Of course, $M$ is already connected, so the analog, if it exists, cannot be that naively straightforward.

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!