Marušič–Scapellato–Zagaglia Salvi compatibility conjecture

Let $\Gamma$ be a graph. Its canonical double cover has adjacency matrix $\begin{pmatrix}0&A\A&0\end{pmatrix}$ when $A$ is the adjacency matrix of $\Gamma$. The matrix $A$ is compatible if there is a nonidentity permutation matrix $P$ of the same size such that $AP$ is the adjacency matrix of some graph. A graph is vertex-determining when distinct vertices have distinct neighborhoods. Marušič–Scapellato–Zagaglia Salvi's compatibility conjecture. A connected non-bipartite vertex-determining graph is unstable if and only if it has a compatible adjacency matrix. The source notes that every nontrivially unstable graph has a compatible adjacency matrix, while the conjecture asserts the converse for connected non-bipartite vertex-determining graphs. The paper's preceding theorem establishes that no arc-transitive nontrivially unstable circulant exists, but does not state a resolution of this broader conjecture.