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Proof of a Brown-Mol conjecture on subtree roots — arXiv:2608.07898

Source: arXiv (en)
Original: https://arxiv.org/abs/2608.07898
License (as archived): CC0-1.0
Archived: 2026-10-19T09:08:48.837Z
SHA-256 of the archived text: c4f7b314af53404e740b390c1fbf0fd61ba12a33ae06b3593a8c7205d1db8366
Stable URL of this page: https://metraipolis.com/documento/dc_edef46cf9105

How to cite

Proof of a Brown-Mol conjecture on subtree roots — arXiv:2608.07898 — arXiv (en), https://arxiv.org/abs/2608.07898 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_edef46cf9105 (Metraipolis archive dc_edef46cf9105, sha256 c4f7b314af53404e740b390c1fbf0fd61ba12a33ae06b3593a8c7205d1db8366)

BibTeX

@misc{dcedef46cf9105,
  title = {Proof of a Brown-Mol conjecture on subtree roots — arXiv:2608.07898},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2608.07898},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_edef46cf9105; sha256 c4f7b314af53404e740b390c1fbf0fd61ba12a33ae06b3593a8c7205d1db8366},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2608.07898. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

The subtree polynomial of a tree is the generating function that enumerates its subtrees according to their orders. Brown and Mol conjectured that every subtree root of a tree of order $n\ge 2$ lies in the disk \[ \left\{z\in\mathbb C: |z|\le 1+\sqrt[n-1]{n-1} \right\}. \] We prove this conjecture by introducing a recursive comparison method based on an extremal problem over integer compositions. We further characterize the equality case: the upper bound is attained if and only if $n$ is even and the tree is the star; in this case the unique boundary root is $-1-\sqrt[n-1]{n-1}$. We also show that every nonzero subtree root $z$ satisfies \[ |z|>\sqrt[n-1]{n-1}-1. \] The lower bound is asymptotically sharp as $n\to\infty$. For odd $n$, although the upper bound is not attained, it is asymptotically sharp as $n\to\infty$.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-08-11 in math.CO).
Categorie arXiv: Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2608.07898 — 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 2026-10-19T09:05:33.544Z dalla pagina https://arxiv.org/catchup/math.CO/2026-08-11?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).