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.



Spectral theory of graph Laplacian besides $\lambda_2$
I came across some exciting references to 'holistic spectral data' while watching [this](https://www.youtube.com/watch?v=8XJes6XFjxM) great talk by prof. James Lee at UWashington, so it might be worth giving a look. The slides are [here](https://simons.berkeley.edu/events/openlectures2014-fall-4). A little search turned up some interesting results:


1. At arount 28:00 in the talk, he mentions that if $\lambda\_i$ and $v\_i$ are the eigenvalues (in increasing order) and eigenvectors of the Laplacian, and $g\_i$ are random $\mathcal{N}(0,1)$ Gaussians, we have
$$\text{cover time of $G$} \asymp |V| \left\|\sum\_{i=2}\frac{g\_i}{\sqrt{\lambda\_i}}v\_i \right\|\_\infty $$
in the sense of tight concentration. Wow! The quantity in the infinity norm, Lee claims, is called the **Gaussian mean field** and is useful in other contexts.
2. At around 30:00 he continues discussing 'holistic spectral data', mentioning how the above spectral characterization helps us prove things about cover time. Examples from electrical networks follow.
3. In the paper <http://arxiv.org/abs/1111.1055>, Lee, Gharan and Trevisan resolve a conjecture saying that there are $k$ eigenvalues close to zero, the graph can be partitioned into $k$ parts with small cuts between them (this was alluded to in the comments).


