Existence of distinct isomorphic switching graphs with incompatible switching data

Construct or rule out distinct isomorphic graphs $\Gamma_1$ and $\Gamma_2$ that have the same set of $Q$-respecting vectors but for which no isomorphism preserves those $Q$-respecting vectors, thereby determining whether the corresponding $(Q,\Gamma_1)$- and $(Q,\Gamma_2)$-switching methods can produce non-isomorphic graphs from the same graph.

Background

The paper discusses whether asymptotic counts for two switching graphs associated with the same orthogonal matrix Q can be added. Non-isomorphic switching graphs can be separated immediately, but distinct isomorphic switching graphs may still define different switching methods if no isomorphism between them fixes the Q-respecting vectors.

Such a pair would affect the aggregation of switching-method counts, because the two methods could produce non-isomorphic cospectral mates from the same input graph. The paper gives no example of this phenomenon and leaves its existence unresolved.

References

If such $\Gamma_1$ and $\Gamma_2$ exist, then the $(Q,\Gamma_1)$- and $(Q,\Gamma_2)$-switching applied to the same graph could produce non-isomorphic graphs. No example of $\Gamma_1$ and $\Gamma_2$ like this is known to the authors.

— Counting cospectral graphs obtained via switching  (2503.08627 - Abiad et al., 11 Mar 2025) in Remark following Lemma 2.2, Section 3.1 “Adding bounds for different switching methods”