Bounds on the smaller eigenvalues of the adjacency matrix of a graph
You might like this recent paper: http://arxiv.org/abs/1211.0589v1

The paper shows, e.g., that "for any finite graph with $n$ vertices and all $k ≥ 2$, the
$k$th largest eigenvalue" of the graph's Laplacian, i.e. $L=I-A/d$, "is at most $1−\Omega(\frac{k^3}{n^3})$", where $A$ is the adjacency matrix and $d$ a bound on the degree.