$\DeclareMathOperator\Coh{Coh}\DeclareMathOperator\Sym{Sym}\DeclareMathOperator\Aut{Aut}$In this post, I asked a question regarding a particular function $\psi$ whose construction is motivated by the graph reconstruction conjecture. I will reiterate the definition here so that this question is self-contained:
Definition of $\psi$
Let $D(G)$ be an unlabelled deck of some simple graph $G$ of order $n$. Choose an enumeration of the subgraphs such that $D(G)=\{G_1,...,G_n\}$. For each $G_i$ we may associate an adjacency matrix of size $(n-1)\times(n-1)$ and we would like to append a row and column of all zeros to these matrices so that they have dimension $n\times n$. We will refer to these matrices as $A_i$ for each $i\in \{1,...,n\}$.
Define,
$$\Psi(S)=\displaystyle\frac{1}{n-2} \displaystyle\sum_{i=1}^{n} S_i A_i S_i^T$$
where $S=(S_1,...,S_n)\in \Sym(n)^n$ and $\Sym(n)$ is the group of $n\times n$ permutation matrices. It can be easily demonstrated that,
- $\mathrm{tr}(\Psi(S))=0$
- $\Psi(S)$ is symmetric
The motivation of $\Psi$ is to encapsulate the reconstruction problem as posed in the motivation section above - the choice of labeling is equivalent to choosing some $n$-tuple of permutations on the "extended" subgraph adjaceny matrices (i.e. $A_i$). We say that $S$ is "coherent" if $\Psi(S)$ is a valid adjacency matrix. Moreover, we denote the set of all coherent values of $S$ as,
$$\Coh(\Psi) = \{S\in \Sym(n)^n \mid \Psi(S)\circ \Psi(S)=\Psi(S)\}$$
where $\circ$ denotes the Hadamard product. It is easy to show that this condition is equivalent to asserting that each entry of $\Psi(S)$ be either $0$ or $1$. This fact coupled with the symmetry and $0$ trace conditions above imply that $\Psi(S)$ would be a valid adjacency matrix.
Description of $\Coh(\psi)$
The underlying algebraic motivation of the definition for $\psi$ is the following observation: If $M$ an $n\times n$ a positive symmetric matrix with $\mathrm{tr}(M)=0$ then,
$$M=\displaystyle\frac{1}{n-2} \displaystyle\sum_{i=1}^{n} P_i M P_i^T$$
where $P_i = I_n - e_i e_i^T$ where $I_n$ is the identity matrix of dimension $n$ and $e_i$ denotes the $i$-th unit vector. Consider a graph $G$ and an associated adjacency matrix $A$. The induced subgraphs of $G$ would have adjacency matrices that are permutation similar to $P_i A P_i^T$ for each $i$. Of course, for Graph Reconstruction, we are given $A_i$ instead, which look like $S_i P_i A P_i^T S_i^T$ for some $S_i \in \Sym(n)$ where $S_i$ is not given.
We know that $|\Coh(\Psi)| \ge 1$ since if $D(G)$ is in fact a deck, then some labeling of the induced subgraph must produce a valid graph. Reconstruction asserts that this graph is in fact unique up to relabeling. Suppose we find one such coherent tuple $S\in \Sym(n)^n$. Consider for each $i$, the set of all $Q\in \Sym(n)$ such that $Q(S_i A_i S_i^T) Q^T = (S_i A_i S_i^T)$, which we will call $\Aut(A_i)$, then clearly $(QS_1,\ldots,QS_n)\in \Coh(\Psi)$.
Empirically, I have found that the following product divides $|\Coh(\Psi)|$ for the case $n=4$,
$$ \displaystyle\prod_{i=1}^n |\Aut(A_i)| $$
Question
Can we fully describe the structure of $\Coh(\Psi)$? Can we find a closed form for $|\Coh(\Psi)|$?
