Asymptotic spectral radius of nonregular graphs

Let ${\lambda }_1(n,\mathrm{\Delta })$ be the maximum adjacency spectral radius among connected nonregular simple graphs of order $n$ and maximum degree $\mathrm{\Delta }$. Using effective resistance bounds, explicit comparison graphs and a one-dimensional Wirtinger inequality, we prove that, for every fixed integer $\mathrm{\Delta }\ge 3$, ${\lambda }_1(n,\mathrm{\Delta })=\mathrm{\Delta }-\frac{c_{\mathrm{\Delta }}{\pi }^2}{4n^2}+O_{\mathrm{\Delta }}(n^{-5\mathrm{/}2})$, where $c_{\mathrm{\Delta }}=\mathrm{\Delta }-1$ for odd $\mathrm{\Delta }$ and $c_{\mathrm{\Delta }}=2(\mathrm{\Delta }-2)$ for even $\mathrm{\Delta }$. This proves the asymptotic conjecture posed by Liu [J. Combin. Theory Ser. B 169 (2024), Conjecture 7.1].