bibkey: suvagiya2026parity authors: Vaibhav Suvagiya year: 2026 title: “Parity families and signed spectra: kernel averaging, near-Ramanujan bounds, and exact circulant models” doi: 10.48550/arXiv.2607.17343 url: https://arxiv.org/html/2607.17343v2 claim: “Conjecture 28 asserts that for every integer m >= 4 the minimum spectral radius over all edge signings of C_(8m)(1,2) is the largest real root of x^4-2x^3-6x^2+12x-4.” strata_touched:

• D5/S3/Combinatorics/Graph/SuvagiyaSignedSquareCycleRefutation license: citation-only triage: anchor

Suvagiya’s period-eight global optimality conjecture

Verified locator

DOI: 10.48550/arXiv.2607.17343

URL: https://arxiv.org/html/2607.17343v2

The source is arXiv:2607.17343v2, section 12.3, Conjecture 28. The withdrawn companion arXiv:2607.18334 is not the source of this assertion.

Statement and conventions

Conjecture 28, titled “Period-8 global optimality”, states:

For every integer m >= 4, min_sigma rho(A_sigma) = r_* on C_(8m)(1,2), where r_* is the largest real root of x^4 - 2x^3 - 6x^2 + 12x - 4 = 0.

The vertices of C_n(1,2) are the residues modulo n. Its undirected edges are {i,i+1} and {i,i+2}; every edge has its own independent sign in {−1,+1}. The adjacency matrix is real and symmetric. Its spectral radius is the maximum absolute value of its eigenvalues. The signings include both possible products of the signs around the Hamilton cycle. Theorem 26 gives an upper bound from a particular periodic family; it does not prove the unrestricted lower bound in Conjecture 28.

Exact counterexample

Take n = 32. Put a_i = +1 for 0 <= i < 31 and a_31 = −1 on {i,i+1}. For {i,i+2}, repeat (1,1,−1,1,−1,−1,1,−1) four times, then replace b_30 by −1 and b_31 by +1. In particular, the three seam edges have weights A_31,0 = −1, A_30,0 = −1, and A_31,1 = +1. The matrix has zero diagonal, 64 undirected edges, four nonzero entries per row, and Hamilton sign product −1.

The squared adjacency is annihilated by

R(y) = y^8−32y^7+416y^6−2816y^5+10568y^4−21632y^3+22168y^2−9408y+1262.

The eight exact finite Horner identities give R(A^2) = 0. All nine coefficients of R((279/100)^2+z) are positive; the constant is 6810961358286782272485104951163521/10^32. Spectral mapping shows that R(lambda^2) = 0 for each real eigenvalue lambda. Since R(y) is positive for y >= (279/100)^2, every absolute eigenvalue is bounded by 279/100. The maximum absolute eigenvalue therefore is at most 279/100. For f(x) = x^4−2x^3−6x^2+12x−4,

f(279/100) = −6766519/100000000 < 0, while f(3) = 5 > 0.

There is a real root strictly between 279/100 and 3. The greatest real root is at least that root, so the exhibited signing has spectral radius strictly less than the conjectured minimum. This refutes the universal assertion by its instance m = 4.

Scope

The unrestricted lower bound is false. Theorem 26’s upper bound is unaffected. The exact optimum at n = 32, the optima at other sizes, and any repaired restriction on the class of signings remain undetermined by this result.

The bounded source and literature screen recorded in issue 11744 found the current author version still presenting Conjecture 28 as open. That screen does not assert exhaustive literature coverage or publication priority.