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Strong permanent inequalities for cycles and a class of single-chord supports

Scott Schubr

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Abstract

For a complex Hermitian positive semidefinite matrix A, write Q(A)=per(A ∘ bar(A)) and d(A)=∏_i a_(ii). We prove Q(A)≤d(A) per(A) whenever the off-diagonal support of A is contained in a cycle of length at least four, or in a theta graph with endpoint-path lengths 1,b,c, where b,c≥3 and b+c≥7. The entries may have arbitrary phases and unequal magnitudes, and singular matrices are included. For a single cycle, the loss between two weighted matching partition functions pays the worst possible directed-cycle contribution. For a single-chord cycle, a simultaneous matching-loss allocation pays all three overlapping cycles. We give the complex form of the known one-point strong-closure argument and obtain hereditary inequalities for graphs assembled from these blocks. These are structural results.

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  1. Edition 2 · 2026-10-27 · public
  2. Edition 1 · 2026-10-22 · public

Research details

Subjects
Mathematics · Linear Algebra and Matrix Inequalities
Keywords
permanent, positive semidefinite matrices, Chollet inequality, cycle support, theta graph, matching partition function
Manuscript date
2026-10-22
Reader reuse
No additional permission granted

Scope

Strong permanent inequalities for complex PSD cycles of length at least four and theta supports with path lengths (1,b,c), b,c ≥ 3 and b+c ≥ 7, together with hereditary closure through the allowed blocks.

Disclosures

AI tools assisted drafting and verification. The author is responsible for the statements, proofs, attribution, and final text. Internal analytic and reader reviews are not external human peer review. Standard cycle-cover expansions, one-point closure, and classical permanent bounds are explicitly credited in the manuscript.

Reproduction

The paper has 7 pages. The editable-source archive contains 4 files. Compile manuscript/main.tex; the analytic proofs and references require no numerical data.

References and related resources

  1. J. Chollet. Is there a permanental analogue to Oppenheim’s inequality? American Mathematical Monthly 89 (1982), 57–58.
  2. S. W. Drury. A counterexample to a question of Bapat and Sunder. Electronic Journal of Linear Algebra 31 (2016), 69–70.
  3. P. Pant and R. Singh. Structural Classes for Chollet’s Permanent Conjecture. arXiv:2604.24192v2 (2026).
  4. R. Mehatari, M. R. Kannan and A. Samanta. On the adjacency matrix of a complex unit gain graph. Linear and Multilinear Algebra 70 (2022), no. 9, 1798–1813.
  5. E. H. Lieb. Proofs of some conjectures on permanents. Journal of Mathematics and Mechanics 16 (1966), 127–134.