Non-star trees with Laplacian peak state transfer

Determine whether infinitely many trees that are not isomorphic to star graphs admit peak state transfer with respect to their Laplacian matrices.

Background

The paper observes that Laplacian peak state transfer is substantially more constrained among trees than adjacency-based peak state transfer. Although every star graph (equivalently, every complete bipartite graph K_{1,n}) exhibits Laplacian peak state transfer, computational results found only one additional example among trees on ten vertices. The unresolved problem asks whether this phenomenon occurs for infinitely many non-star trees.

References

Determine whether infinitely many such trees, not isomorphic to the star graph, admit Laplacian peak state transfer.

— Peak state transfer in continuous quantum walks  (2505.11986 - Coutinho et al., 17 May 2025) in Section 6, Conclusions and open problems, immediately following the sentence “A natural open problem is the following:”