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).

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).

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).

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).