Extension to composite orders for strongly regular graphs

Determine whether the quantum walk characteristic polynomial χ_q(G, λ) distinguishes strongly regular graphs of composite order n, for example n = pq or n = p^2, up to isomorphism, thereby extending the prime-order result.

Background

The main theorem proves that for strongly regular graphs on a prime number of vertices (with degree k ≥ 6), the quantum walk characteristic polynomial completely determines the graph up to isomorphism. The proof critically exploits the prime-order structure via the discrete Fourier transform over Z_p and a factor-by-factor argument relying on unique factorization.

For composite orders, these structural tools do not directly apply, and even seemingly modest extensions such as n = pq or n = p2 would require new ideas.

References

Several directions remain open and appear tractable. The most natural extension concerns composite orders: the block decomposition exploits primality in a fundamental way, through both the orthogonality of characters of $Z_p$ and the unique factorization step in the proof of the main theorem, so new ideas would be required even for $n = pq$ or $n = p2$.

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