Strong permanent inequalities on three chorded seven-cycle supports
Loading original first page…
Abstract
We prove per(|A|^2)≤per(A) for complex positive semidefinite correlation matrices supported on a seven-cycle with chords 03,15,26. More precisely, the difference dominates one fiftieth of the matching-partition loss. The proof combines elementary cycle estimates with an explicit nonnegative integer polynomial identity, supplied with a standard-library verification program. The two-chord subgraph with chords 03,15 inherits the same bound. For the crossing placement 03,14 we give a separate analytic proof with constant 1/12. All results include missing edges and singular matrices. The normalized arguments require only vertex-star magnitude bounds, and we describe equality and diagonal scaling.
Read the full paper in the original PDF.
Edition files
- Strong_Permanent_Multichord_Seven_Cycles.pdf94.8 KiB · pdf
SHA-256
076ed867fae181ad7e1fcd99abc56c93c959b8f2ff6b5a5fdae24b4eb52dcd88 - Revised_publication_package.zip106.0 KiB · package
SHA-256
c9e3411a3e8d1782f8d6cfa9e136314cbb55c90db419f24064b2666185a7c463
Available editions
Research details
- Subjects
- Mathematics · Linear Algebra and Matrix Inequalities
- Keywords
- permanent inequality, positive semidefinite matrix, Chollet conjecture, seven-cycle support, graph matching, exact integer certificate
- Manuscript date
- 2026-10-22
- Reader reuse
- No additional permission granted
Scope
Quantitative strong permanent inequalities on the three specified chorded seven-cycle supports and their subgraphs, with matching-loss factors 1/50 or 1/12 under the stated vertex-star magnitude bounds.
Disclosures
AI tools assisted drafting and verification, as disclosed in the manuscript; the author is responsible for the final statements, proofs and attribution. Internal analytic and exact-code reviews are not external human refereeing. Originality remains provisional. The preceding cycle/single-chord paper, structural graph literature, cycle expansion and polynomial-positivity framework are credited.
Reproduction
The paper has 7 pages. The editable-source archive contains 6 files. Compile manuscript/main.tex. The standard-library verify_certificate.py reconstructs the 323-term integer identity and graph budgets.
References and related resources
- J. Chollet, Is there a permanental analogue to Oppenheim’s inequality?, American Mathematical Monthly 89 (1982), 57–58
- S. W. Drury, A counterexample to a question of Bapat and Sunder (2016)
- P. Pant and R. Singh, Structural Classes for Chollet’s Permanent Conjecture, arXiv:2604.24192v2
- S. Schubr, Strong permanent inequalities for complex positive semidefinite cycle and single-chord supports, edition 1 (2026)
- D. Handelman, Representing polynomials by positive linear functions on compact convex polyhedra (1988)
- R. Mehatari, M. R. Kannan and A. Samanta, On the adjacency matrix of a complex unit gain graph (2022)