Extremal spectral radius of nonregular graphs with prescribed maximum degree

Let G be a graph attaining the maximum spectral radius among all connected nonregular graphs of order n with maximum degree Δ. Let ${\lambda }_1\left(G\right)$ be the spectral radius of G. A nice conjecture due to Liu et al. (2007) [19] asserts that$\underset{n\to \infty }{\lim }\frac{n^2(\Delta -{\lambda }_1(G\left)\right)}{\Delta -1}={\pi }^2$ for each fixed Δ. Concerning an important structural property of the extremal graphs G, Liu and Li (2008) [17] put forward another conjecture which states that G has exactly one vertex of degree strictly less than Δ. In this paper, we make progress on the two conjectures. To be precise, we disprove the first conjecture for all $\Delta \ge 3$ by showing that$\underset{n\to \infty }{\lim \sup }\frac{n^2(\Delta -{\lambda }_1(G\left)\right)}{\Delta -1}\le \frac{{\pi }^2}{2}.$ For small Δ, we determine the precise asymptotic behavior of $\Delta -{\lambda }_1\left(G\right)$. In particular, we show that $\underset{n\to \infty }{\lim }{n}^2(\Delta -{\lambda }_1(G\left)\right)/(\Delta -1)={\pi }^2/4$ if $\Delta =3$; and $\underset{n\to \infty }{\lim }{n}^2(\Delta -{\lambda }_1(G\left)\right)/(\Delta -2)={\pi }^2/2$ if $\Delta =4$. We also confirm the second conjecture for $\Delta =3$ and $\Delta =4$ by determining the precise structure of extremal graphs. Furthermore, we show that the extremal graphs for $\Delta \in \{3,4\}$ must have a path-like structure built from specific blocks.