Sharper bounds for strongly cospectral pairs without perfect state transfer

Determine whether tighter upper bounds can be established for the maximum state-transfer amplitude or probability between strongly cospectral vertex pairs in graphs that do not admit perfect state transfer.

Background

For strongly cospectral vertices u and v, the paper notes that the bounding-matrix entry B(A)(u,v) equals 1, so peak state transfer is equivalent to perfect state transfer. Many graphs nevertheless contain strongly cospectral vertex pairs for which perfect state transfer does not occur. The authors explicitly leave unresolved whether the maximum achievable state transfer for such pairs admits upper bounds sharper than the general bound supplied by the bounding matrix.

References

There are many graphs with strongly cospectral pairs of vertices that do not admit perfect state transfer. In this case, one can ask if there are tighter upper bounds on the maximum state transfer.

— Peak state transfer in continuous quantum walks  (2505.11986 - Coutinho et al., 17 May 2025) in Section 6, Conclusions and open problems, final paragraph before the acknowledgements