Assume you have a graph with an equitable partition with respect to cells $V_1,\ldots,V_n$. Accordingly, you can take the cellwise average value of a function on the node set - in other words, the projection of the node space onto the susbpace of cellwise constant functions. Let us call $P$ this projector.

Then, it is well-known (see e.g. Bollobas' book, Prop. VIII.3.15; or Brouwer-Haemer's book, §2.3) that there is a matrix $C$ such that $P$ intertwines with $A$ and $C$, i.e., $AP=PC$, where $A$ is the adjacency matrix of the graph; in fact, $C$ can be investigated as the adjacency matrix of a certain auxiliary "quotient" graph, with certain nice connections between the spectra of $A$ and $C$.

Now, what I'd like to know is what happens if we consider $(I-P)$ instead of $P$, or - if you prefer - the projector onto the null space of $P$, instead of its range. Is there a matrix $D$ such that $(I-P)$ intertwines with $A$ and $D$? Can $D$ be interpreted as the adjacency matrix of a certain auxiliary graph, again?

(If necessary, in the above question you can gladly replace the adjacency matrix by the discrete laplacian, the normalized laplacian, or the signless laplacian).