bibkey: malkoun2025finitegraphs
authors: Joseph Malkoun
year: 2025
title: "Finite graphs and configurations of points"
doi: 10.1016/j.geomphys.2026.105993
url: https://arxiv.org/abs/2508.13472v3
claim: "The paper generalizes the Atiyah problem on configurations of points by graph amplitudes and states Conjecture D: for two equivalence relations on [n] with class-preserving permutation groups P_1, P_2 and every Hermitian positive semidefinite H, Re f(H) >= |P_1 ∩ P_2| prod_i H_ii, where f(H) sums prod_i H_{sigma_1(i), sigma_2(i)} over P_1 × P_2; it proves the case of nested relations and that Conjecture D implies its tree Conjecture C."
strata_touched:
  - D5/S3/Quantum/Matrix/MalkounYoungSubgroupConjectureRefutation
license: citation-only
triage: anchor

Malkoun, Finite graphs and configurations of points

J. Malkoun, arXiv:2508.13472v3 [math-ph] (7 September 2026; v1 19 August 2025, v2 9 March 2026); Journal of Geometry and Physics, DOI 10.1016/j.geomphys.2026.105993 (Crossref: title, author and journal match; issue date January 2027). Quotations are from the v3 TeX source, with the macro \Real written as $\mathrm{Re}$. The journal text was not read.

The groups and the function (Section “A conjecture in matrix analysis”):

Given an equivalence relation, say $\sim$, on $[n]$, we let $P_{\sim }$ be the group of all permutations of $[n]$ which preserve the equivalence classes of $[n]$ under $\sim$. Let ${\sim }_i$, for $i=1,2$, be two equivalence relations on $[n]=\{1,\dots ,n\}$, with associated permutation groups $P_1$, $P_2$, respectively

Let $H$ be a Hermitian positive semidefinite $n\times n$ matrix. We define $f_{{\sim }_1,{\sim }_2}(H)={\sum }_{{\sigma }_1\in P_1}{\sum }_{{\sigma }_2\in P_2}{\prod }_{i=1}^n{H}_{{\sigma }_1(i),{\sigma }_2(i)}.$

Conjecture D:

Under the above hypotheses, we have $\mathrm{Re}(f_{{\sim }_1,{\sim }_2}(H))\ge |P_1\cap P_2|{\prod }_{i=1}^n{H}_{ii}.$

The source records that the discrete and indiscrete relations give Marcus’s permanent analogue of the Hadamard inequality, proves Conjecture D when one relation is finer than the other, proves that Conjecture D implies its Conjecture C for trees, and reports that numerical simulations for small $n$ found no counterexample. The same inequality is MathOverflow question 480580 (Malkoun, 13 October 2024), which proves the case $n=3$ with classes $\{1,2\},\{3\}$ and $\{1\},\{2,3\}$.

Verified locator

DOI: https://doi.org/10.1016/j.geomphys.2026.105993 (Journal of Geometry and Physics; the journal text was not read).

Source: https://arxiv.org/abs/2508.13472v3

• URL: https://arxiv.org/abs/2508.13472v3 (the e-print is a tar archive; member finite_graphs_and_configurations_of_points_v5.tex, 842 lines, md5 3c520a352e066a90cdcffb75a2ee568c):
  • the groups and the function, l. 405–408;
  • Conjecture D, l. 410–412;
  • the Marcus case, l. 414;
  • Conjecture D implies Conjecture C, Theorem, l. 416;
  • the nested case, Proposition, l. 473;
  • the numerical report, l. 744.
• The author, the version dates and the journal DOI are from the arXiv abstract page and Crossref.