Non-vertex-transitive strongly regular graphs

Determine whether the quantum walk characteristic polynomial χ_q(G, λ) is a complete isomorphism invariant for strongly regular graphs that are not vertex-transitive.

Background

The proof technique in the paper leverages the Cayley (circulant) structure available for prime-order graphs, which are vertex-transitive, to obtain a Fourier block decomposition and recover the connection set.

For strongly regular graphs that are not vertex-transitive, this structural leverage is absent, and it is unclear whether a different argument can establish the same completeness of χ_q.

References

A related and more ambitious question is whether $\chi_q$ is a complete isomorphism invariant for strongly regular graphs that are not vertex-transitive: our numerical experiments found no counterexample among non-isomorphic graphs on $n \leq 10$ vertices, but the algebraic mechanism used here relies on the Cayley structure in an essential way, and it is unclear whether a substitute argument exists in the general setting.

— The Quantum Walk Characteristic Polynomial Distinguishes All Strongly Regular Graphs of Prime Orde  (2604.01507 - Roldan, 2 Apr 2026) in Section 7 (Concluding Remarks)