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arXiv · 2610.04947

Asymptotic spectral radius of nonregular graphs

Hangxi Cha·Haiying Shan
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Abstract

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 $Δ\ge3$, $λ_1(n,Δ)=Δ-\frac{c_Δπ^2}{4n^2}+O_Δ(n^{-5/2})$, 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].

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Hangxi Cha, Haiying Shan. 2026-10-04. Asymptotic spectral radius of nonregular graphs. https://arxiv.org/abs/2610.04947

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