Lower bound on the number of edges for peak state transfer

Find a lower bound, in terms of the distance d between vertices u and v, for the number m of edges in a graph G that admits peak state transfer between u and v at distance d.

Background

The paper seeks to understand how sparse a graph can be while supporting peak state transfer over a prescribed distance. For a graph G with n vertices, m edges, and vertices u and v at distance d that admit peak state transfer, the authors ask for a lower bound on m depending on d. This is presented as the peak-state-transfer analogue of a previously posed problem for perfect state transfer.

References

Another natural question concerns the relationship between graph sparsity and the ability to support peak state transfer. Suppose we have a graph $G$ on $n$ vertices with $m$ edges and peak state transfer between $u,v$ at distance $d$. How small can $m$ be? More formally, we have the following:

— 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 discussion of graph sparsity

If $G$ admits peak state transfer between $u,v$ at distance $d$ and has $m$ edges, find a lower bound for $m$ in terms of $d$.

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