Assume you have a graph with an [equitable partition](https://mathoverflow.net/questions/96858/complexity-of-equitable-partitions) 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).



a variation on the theory of equitable partitions for graphs
The column space of $I-P$ is $A$-invariant and therefore there is a matrix $D$ such that
$A(I-P)=(I-P)D$. The difficulty is that $D$ will generally not be non-negative and so
it it less useful to interpret it as a weighted adjacency matrix. Additionally, in 
practice $C$ is small (which is what makes it useful) and so the size of $D$ will be
comparable with that of $A$. In which case we might as well work directly with $A$.


On the other hand, if the cells of the partition all have size two, then $D$ is a signed adjacency matrix, and so in this case we can have some fun.


