bibkey: polettini2024coplanarity
authors: Matteo Polettini; Pedro E. Harunari; Sara Dal Cengio; Vivien Lecomte
year: 2024
title: "Coplanarity of rooted spanning-tree vectors"
doi: 10.1007/s11005-025-02026-8
url: https://arxiv.org/abs/2407.16093v1
claim: "The authors conjecture that the 3ⁿ deletion-constriction vectors through n reversible edges span a space of dimension n+1."
strata_touched:
  - D5/S3/StatisticalMechanics/PolettiniPinnedTreeSpanRefutation
license: citation-only
triage: anchor

Coplanarity of rooted spanning-tree vectors

M. Polettini, P. E. Harunari, S. Dal Cengio and V. Lecomte, Coplanarity of rooted spanning-tree vectors, arXiv:2407.16093v1; Letters in Mathematical Physics (2025).

Verified locator

DOI: https://doi.org/10.1007/s11005-025-02026-8

Source: https://arxiv.org/abs/2407.16093v1

The arXiv v1 PDF supplies the locators below. The journal full text is not checked.

Source statements

Page 2, Introduction: “Finally, we conjecture that the 3ⁿ-vectors representing the deletion-constriction through n reversible edges span a n+1 dimensional space.”

Page 6, §2.1: “We assume the graph to be irreducible, that is, there exists a path x ⇝ y of nonzero weight between any two vertices.”

Page 6, §2.1: “In a directed graph, a rooted spanning tree Tₓ with root x is a spanning tree such that each edge is directed along the unique path that leads to the root.”

Page 6, Definition 1: “The conditioned rooted spanning tree polynomial is defined as”

[ \tau_x^{\setminus A,\setminus!\setminus B} =\sum_{T_x:\ A\cap T_x=\varnothing,\ B\subseteq T_x}r(T_x). ]

Page 17, Conclusions: “Given the cases n = 1 and n = 2, it is tempting to speculate that the dimension of the linear space spanned by them is n+1. We collected numerical evidence supporting this conjecture in the case n = 3 (in which case the spanning-tree vectors have 27 entries), for graphs up to 13 vertices and randomized values of the weights of the unpinned edges.”

Mathematical correspondence

The vertices are Fin N, and r x y > 0 specifies the directed edges. A rooted tree is a parent map that fixes its root, has a positive-rate outgoing edge at every other vertex, and reaches the root by iteration from every vertex. Its weight is the product over the non-root vertices. The root loop is auxiliary and is excluded from the product. Each pin has status none, some true or some false, for neither direction, the forward direction only or the backward direction only. The sum defining tau is the conditioned polynomial for that status, without rescaling its edge weights.

The universal dimension statement includes nonnegative rates, irreducibility, distinct reversible pins and irreducibility after deleting the pins. Unit-rate K4 with the triangle pinned satisfies all these hypotheses but has dependent root vectors.

Experiment reference

The exact finite computation is in the-omega-institute/trureturing-experiments, docs/reports/polettini-harunari-dalcengio-lecomte-2024-pinned-cycle-rank/, commit f7b148864def985e33d70c594831b800ba27c22e. The check.py SHA-256 is 104a55a023e87413b7aa73a9a40192a7ec445619a06f9bedd0e8f1e1444ea1cd; the source preregistration reports python3 check.py, exit 0, RESULT PASS.

Literature boundary

Literature completeness is ASSUMED-UNVERIFIED. The source is the pinned arXiv v1; no general incidence-rank formula is asserted here.