Most of what I've seen about the spectral theory of the graph Laplacian concentrates on $\lambda_2$, the second-smallest eigenvalue. This eigenvalue contains information regarding the connectivity of the graph.

What can one learn from the rest of the eigenvalues and their associated eigenvectors? I'm not interested in special graphs -- e.g. regular graphs -- but large, messy graphs created from data.

Statistics and Computing, vol. 17, no. 4? If I recall (though I don't have it handy), there is at least one chapter of D. Skillicorn,Understanding Complex Datasets: Data Mining with Matrix Decompositions, Chapman and Hall/CRC, 2007 that discusses some high-level aspects, as well. $\endgroup$ – cardinal Jul 15 '13 at 18:506more comments