Intuition for Laplacian matrix of a graph's eigenvectors and eigenvalues I am having difficulty finding intuition for Laplacian matrix eigenvalues/vectors in terms of non-regular, non-complete graphs. For example, consider the L, Laplacian, on a graph, G, a set of points in R. Consider it's 3rd largest eigenvalue, what intuition can I derive from it and it's unique eigenvector? 
Also, what happens to the rowspace/columnspace? The proofs I've come across are elaborate and use Rayleigh quotients, but I have no interest in elaborate proofs but rather real intuition.
 A: To address your first question (in a non-rigorous, hand-wavey, but hopefully intuitive manner):
Forget about graphs for a moment, and consider periodic scalar functions defined on $\mathbb {R}$.  Each of these functions can be expressed as a Fourier series (a linear combination of an orthonormal set of basis sine and cosine functions).
It turns out that Fourier series generalizes to compact Riemannian manifolds, where we can represent square-integrable functions in an analogous way.  However, in this setting, the orthonormal basis set for square-integrable functions on a given manifold turns out to be the eigenfunctions of its Laplace-Beltrami operator.  For a concrete example of this, take a look at Laplace spherical harmonics.
Now, to leap from compact Riemannian manifolds to graphs, Belkin and Niyogi show that, "under certain conditions, eigenvectors of the graph Laplacian converge to eigenfunction [sic] of the Laplace-Beltrami operator on the underlying manifold."  Their analysis deals with graphs derived from point cloud data, so they can (a) assume that each point was sampled from the underlying manifold, and (b) they have a natural geometric way to build a suitable graph Laplacian (a so-called point cloud Laplacian) that effectively approximates the Laplace-Beltrami operator of the underlying, unknown manifold.  I'm not sure about how (or if it is even possible) to generalize their dependence on point cloud data, but maybe a metric-preserving embedding could be used as an implicit point cloud for an arbitrary graph Laplacian.
So, you can think of the eigenvectors of a graph Laplacian as a sort of Fourier basis for scalar-valued functions defined on the nodes of the graph.  The increasing magnitudes of the eigenvalues correspond to increasing frequencies of the eigenvectors.
So, to understand the 3rd largest eigenvalue, you could imagine that, if you were to define some function on the nodes of your graph, then decompose it using the Laplacian eigenvectors as a basis, you would find that the 3rd largest eigenvalue would capture higher-frequency features than the eigenvectors with smaller eigenvalues.  You would also notice that the 2nd- and 1st-largest eigenvectors would capture even higher-frequency features.  
For a concrete example of this in $\mathbb{R^3}$, see this paper by Olga Sorkine.
