Perfect state transfer on unicyclic graphs

@misc{song2026perfect,
  title = {{Perfect state transfer on unicyclic graphs}},
  author = {Xingkun Song},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.41572},
  eprint = {2610.41572},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.41572},
}

TY  - UNPB
TI  - Perfect state transfer on unicyclic graphs
AU  - Xingkun Song
PY  - 2026
DA  - 2026/10/30
PB  - arXiv
DO  - 10.48550/arxiv.2610.41572
UR  - https://arxiv.org/abs/2610.41572
AB  - We characterize adjacency perfect state transfer in connected unicyclic graphs: the transfer vertices are opposite degree-two vertices of a four-cycle, and the two attached trees satisfy a characteristic-polynomial condition. Except for the four-cycle, the transfer pair is unique and the minimum transfer time is π/√2. Outside the four-cycle, we obtain sharp lower bounds of $46$ on the order and $10$ on the maximum degree, and classify all examples of maximum degree at most $20$. A finite inverse construction enumerates all realizations of a prescribed eigenvalue support at the transfer vertices. For supports of size at least seven, we establish the sharp maximum-degree lower bound of $21$. Every seven-element support {0,±√2,± u√2,± v√2}, with odd integers 3≤ u<v, is realizable, and every such realization has attachment depth at most four. Explicit branching rules give a minimum-order recurrence. Among all connected unicyclic graphs admitting perfect state transfer with seven-element eigenvalue support, the minimum order is $218$, attained uniquely up to isomorphism.
ER  -