Asymptotic spectral radius of nonregular graphs

Abstract

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

11 pages, 2 figures