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A spectral Moore problem and the Yang-Koolen conjecture
Abstract
We study the maximum order v(k,θ) of a connected k-regular graph whose second largest eigenvalue is at most θ. For every real θ≥1, we prove that T(θ) =limsupk→∞(v(k,θ)−2k) =2⌊1+θ2/4⌋. In particular, T(θ)>2θ for every real θ≥4, disproving the Yang-Koolen conjecture. We also determine v(k,θ) for all sufficiently large k on explicit intervals of θ. We prove a jump in the second largest eigenvalue above $1$. For each fixed 1≤θ<2 and all sufficiently large k, every connected k-regular graph of order at least 2k+2 whose second largest eigenvalue is at most θ is the crown graph. For θ=√2, we prove that v(k,√2)=2k+2 for every k≥13. Finally, we obtain two new upper bounds on v(k,θ).
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BibTeX
@misc{gupta2026spectral,
title = {{A spectral Moore problem and the Yang-Koolen conjecture}},
author = {Vishal Gupta},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00591},
eprint = {2611.00591},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00591},
}
RIS
TY - UNPB
TI - A spectral Moore problem and the Yang-Koolen conjecture
AU - Vishal Gupta
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00591
UR - https://arxiv.org/abs/2611.00591
AB - We study the maximum order v(k,θ) of a connected k-regular graph whose second largest eigenvalue is at most θ. For every real θ≥1, we prove that T(θ) =limsup_k→∞(v(k,θ)−2k) =2⌊1+θ^2/4⌋. In particular, T(θ)>2θ for every real θ≥4, disproving the Yang-Koolen conjecture. We also determine v(k,θ) for all sufficiently large k on explicit intervals of θ. We prove a jump in the second largest eigenvalue above $1$. For each fixed 1≤θ<2 and all sufficiently large k, every connected k-regular graph of order at least 2k+2 whose second largest eigenvalue is at most θ is the crown graph. For θ=√2, we prove that v(k,√2)=2k+2 for every k≥13. Finally, we obtain two new upper bounds on v(k,θ).
ER -