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Perfect state transfer on unicyclic graphs — arXiv:2610.41572

Source: arXiv (en)
Original: https://arxiv.org/abs/2610.41572
License (as archived): CC0-1.0
Archived: 2026-11-02T05:00:37.633Z
SHA-256 of the archived text: efb7537851a8dcb3727c226e0683cf93e44ad540d3f8d2a533820b8db7764a10
Stable URL of this page: https://metraipolis.com/documento/dc_f23ebbc6dfeb

How to cite

Perfect state transfer on unicyclic graphs — arXiv:2610.41572 — arXiv (en), https://arxiv.org/abs/2610.41572 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_f23ebbc6dfeb (Metraipolis archive dc_f23ebbc6dfeb, sha256 efb7537851a8dcb3727c226e0683cf93e44ad540d3f8d2a533820b8db7764a10)

BibTeX

@misc{dcf23ebbc6dfeb,
  title = {Perfect state transfer on unicyclic graphs — arXiv:2610.41572},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2610.41572},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_f23ebbc6dfeb; sha256 efb7537851a8dcb3727c226e0683cf93e44ad540d3f8d2a533820b8db7764a10},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2610.41572. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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 $\pi/\sqrt2$. 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 $\{0,\pm\sqrt2,\pm u\sqrt2,\pm v\sqrt2\}$, with odd integers $3\leq u<v$, 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.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv math.CO, nuovi lavori del giorno).
Categorie arXiv: Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2610.41572 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-11-02T04:47:57.688Z dalla pagina https://arxiv.org/list/math.CO/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).