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Asymptotic maximum spectral radius of nonregular graphs
Abstract
Let λ1(n,Δ) be the maximum adjacency spectral radius among connected nonregular simple graphs of order n and maximum degree Δ. Using effective resistance bounds, explicit comparison graphs and a one-dimensional Wirtinger inequality, we prove that, for every fixed integer Δ≥3, λ1(n,Δ)=Δ−cΔπ2/(4n2)+OΔ(n−5/2) as n→∞, where cΔ=Δ−1 for odd Δ and cΔ=2(Δ−2) for even Δ. This proves the asymptotic conjecture posed by Liu [J. Combin. Theory Ser. B 169 (2024), Conjecture 7.1].
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BibTeX
@misc{cha2026asymptotic,
title = {{Asymptotic maximum spectral radius of nonregular graphs}},
author = {Hangxi Cha and Haiying Shan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04947},
eprint = {2610.04947},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04947},
}
RIS
TY - UNPB
TI - Asymptotic maximum spectral radius of nonregular graphs
AU - Hangxi Cha
AU - Haiying Shan
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.04947
UR - https://arxiv.org/abs/2610.04947
AB - Let λ_1(n,Δ) be the maximum adjacency spectral radius among connected nonregular simple graphs of order n and maximum degree Δ. Using effective resistance bounds, explicit comparison graphs and a one-dimensional Wirtinger inequality, we prove that, for every fixed integer Δ≥3, λ_1(n,Δ)=Δ−c_Δπ^2/(4n^2)+O_Δ(n^−5/2) as n→∞, where c_Δ=Δ−1 for odd Δ and c_Δ=2(Δ−2) for even Δ. This proves the asymptotic conjecture posed by Liu [J. Combin. Theory Ser. B 169 (2024), Conjecture 7.1].
ER -