Strong permanent inequalities for complex positive semidefinite cycle and single-chord supports
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Abstract
For a complex Hermitian positive semidefinite matrix A, write Q(A) = per(A ◦ Ā) and d(A) = ∏ᵢ aᵢᵢ. 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 for the stated permanent comparisons.
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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
The strong self-conjugate inequality is proved for complex Hermitian positive semidefinite matrices supported on cycles of length at least four and theta graphs with path lengths 1, b, c, where b,c ≥ 3 and b+c ≥ 7, together with the stated hereditary block closure. Arbitrary phases, unequal magnitudes, singular matrices, and missing edges are included. The paper excludes theta(1,3,3), further chords, unrestricted Chollet, and arbitrary-second-factor Bapat–Sunder conclusions. Paired consequences require both factors to meet the proved structural hypotheses.
Disclosures
AI tools assisted drafting and verification. The author is responsible for the statements, proofs, attribution, and final text. Standard cycle-cover expansions, one-point closure, and classical permanent bounds are explicitly credited in the manuscript.
Reproduction
The four-file package contains the complete seven-page manuscript PDF, editable LaTeX source, reader instructions, and a SHA-256 manifest. The proof is analytic and self-contained apart from the credited standard inequalities and structural predecessors. Compile manuscript/main.tex with a standard LaTeX installation; the package README provides the reader instructions.
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. Electronic Journal of Linear Algebra 31 (2016), 69–70.
- P. Pant and R. Singh. Structural Classes for Chollet’s Permanent Conjecture. arXiv:2604.24192v2 (2026).
- 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.
- E. H. Lieb. Proofs of some conjectures on permanents. Journal of Mathematics and Mechanics 16 (1966), 127–134.