Parity families and signed spectra: kernel averaging, near-Ramanujan bounds, and exact circulant models
Abstract
We develop an affine 𝔽2 framework for structured signings of regular graphs. A family-averaging identity converts even spectral moments into parity-weighted closed-walk counts supported on the span of prescribed short even cycles, while a kernel-averaged Ihara identity gives the corresponding decomposition at the non-backtracking level. We give a finite-scale bounded-rank counting estimate and a conditioning corollary showing that, on bicycle-free graph sequences, any parity family of uniformly bounded codimension contains near-Ramanujan signings whenever the corresponding random-signing theorem applies. The latter is a transfer statement rather than a new concentration theorem. Finally, on Cn(1,2) for even n≥10, the quadrilateral-unbalanced family has exactly four switching classes and its twisted classes attain ρ-(n)=2√(cos2(π/n)+cos2(2π/n)); a period-$8$ signing has spectral radius r*=2.793604493334841… for every positive multiple of $8$. Thus for n=8m≥32 the constrained minimum is strictly larger than a value attained by an unrestricted signing, while equality of r* with the unrestricted minimum remains conjectural.
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BibTeX
@misc{suvagiya2026parity,
title = {{Parity families and signed spectra: kernel averaging, near-Ramanujan bounds, and exact circulant models}},
author = {Vaibhav Suvagiya},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.17343},
eprint = {2607.17343},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.17343},
}
RIS
TY - UNPB
TI - Parity families and signed spectra: kernel averaging, near-Ramanujan bounds, and exact circulant models
AU - Vaibhav Suvagiya
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2607.17343
UR - https://arxiv.org/abs/2607.17343
AB - We develop an affine 𝔽_2 framework for structured signings of regular graphs. A family-averaging identity converts even spectral moments into parity-weighted closed-walk counts supported on the span of prescribed short even cycles, while a kernel-averaged Ihara identity gives the corresponding decomposition at the non-backtracking level. We give a finite-scale bounded-rank counting estimate and a conditioning corollary showing that, on bicycle-free graph sequences, any parity family of uniformly bounded codimension contains near-Ramanujan signings whenever the corresponding random-signing theorem applies. The latter is a transfer statement rather than a new concentration theorem. Finally, on C_n(1,2) for even n≥10, the quadrilateral-unbalanced family has exactly four switching classes and its twisted classes attain ρ_-(n)=2√(cos^2(π/n)+cos^2(2π/n)); a period-$8$ signing has spectral radius r_*=2.793604493334841… for every positive multiple of $8$. Thus for n=8m≥32 the constrained minimum is strictly larger than a value attained by an unrestricted signing, while equality of r_* with the unrestricted minimum remains conjectural.
ER -