If a graph $G$ has an equitable partition, then its charachteristic polynomial (for the adjacency matrix) has a divisor that can be seen as the characteristic polynomial (for the adjacency matrix) of a oriented and weighted, but smaller, graph $G'$. In other words, the spectrum of $G$ contains the spectrum of $G'$. Equality (without counting multiplicity) holds if the graph has further properties, in particular if it is walk regular (an old result by Godsil and McKay).

If $G$ only has an almost equitable partition, then the above assertion still holds if we replace "adjacency matrix" by "discrete laplacian". Is it known whether equality of spectra holds in special cases?