In the context of Szemeredi's regularity lemma, is there any way to relate the eigenvalues of the ambient graph with the densities of an $\epsilon$-regular partition?
More precisely, if $V(G) = V_{0} \cup V_{1} \cup \ldots \cup V_{k}$ is an $\epsilon$-regular partition for the vertex set of a graph $G$ (with exeptional set $V_{0}$), and if $Q=(d_{ij})_{1\leq i,j \leq k}$ is the density matrix of the partition (i.e. $d_{ij}$ is the edge density between $V_{i}$ and $V_{j}$), is it true that the eigenvalues of $G$ are close to the eigenvalues of $Q$ (times some constant)?
I would be very interested in something like this. For a reminder of the definitions of edge density and an $\epsilon$-regular partition (as as the Regularity Lemma), see http://en.wikipedia.org/wiki/Szemer%C3%A9di_regularity_lemma.