Signed circulants at the Ramanujan bound
Abstract
For the circulant graph Cn(1,2) with n≥10 even, the F2 system requiring every quadrilateral to be unbalanced is consistent and its solutions form exactly four switching classes. We show that the class containing the signing which is +1 on step-$1$ edges and (−1)i on step-$2$ edges has spectrum {±2√(cos2θk+cos2 2θk)} and spectral radius exactly 2√2, well below the Kesten bound 2√3; that the quadrilateral system is equivalent to alternating triangle fluxes, so that the four classes are coordinatized by (τ0,α) and the spectral radius depends only on the Hamilton-cycle holonomy α; and that the two twisted classes attain ρ-(n)=2√(cos2(π/n)+cos2(2π/n))<2√2. Exhaustive enumeration of all 2n+1 switching classes for n∈{8,10,12,14,16,18} shows that ρ-(n) is the global minimum in every case, and we conjecture this for all even n; the lower bound is a flux-minimization statement in the sense of Lieb's flux-phase theorem. For odd n the quadrilateral system is inconsistent.
Related
- Parity families and signed spectra: kernel averaging, near-Ramanujan bounds, and exact circulant models
- The strong Ramanujan bound for Drinfeld modular forms
- An upper bound on the smallest singular value of dense signed random combinatorial matrices
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- Two-basepoint Terwilliger algebras and the quantum symmetry of prime-order circulants
BibTeX
@misc{suvagiya2026signed,
title = {{Signed circulants at the Ramanujan bound}},
author = {Vaibhav Suvagiya},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.18334},
eprint = {2607.18334},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.18334},
}
RIS
TY - UNPB
TI - Signed circulants at the Ramanujan bound
AU - Vaibhav Suvagiya
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2607.18334
UR - https://arxiv.org/abs/2607.18334
AB - For the circulant graph C_n(1,2) with n≥10 even, the F_2 system requiring every quadrilateral to be unbalanced is consistent and its solutions form exactly four switching classes. We show that the class containing the signing which is +1 on step-$1$ edges and (−1)^i on step-$2$ edges has spectrum {±2√(cos^2θ_k+cos^2 2θ_k)} and spectral radius exactly 2√2, well below the Kesten bound 2√3; that the quadrilateral system is equivalent to alternating triangle fluxes, so that the four classes are coordinatized by (τ_0,α) and the spectral radius depends only on the Hamilton-cycle holonomy α; and that the two twisted classes attain ρ_-(n)=2√(cos^2(π/n)+cos^2(2π/n))<2√2. Exhaustive enumeration of all 2^n+1 switching classes for n∈{8,10,12,14,16,18} shows that ρ_-(n) is the global minimum in every case, and we conjecture this for all even n; the lower bound is a flux-minimization statement in the sense of Lieb's flux-phase theorem. For odd n the quadrilateral system is inconsistent.
ER -