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 $\mathrm{\Delta }$. Let ${\lambda }_1(G)$ be the spectral radius of $G$. A nice conjecture due to Liu, Shen and Wang [On the largest eigenvalue of non-regular graphs, J. Combin. Theory Ser. B, 97 (2007) 1010–1018] asserts that $$\underset{n\to \mathrm{\infty }}{\lim }\frac{n^2(\mathrm{\Delta }-{\lambda }_1(G))}{\mathrm{\Delta }-1}={\pi }^2$$ for each fixed $\mathrm{\Delta }$. Concerning an important structural property of the extremal graphs $G$, Liu and Li present another conjecture which states that $G$ has degree sequence $\mathrm{\Delta },\dots ,\mathrm{\Delta },\delta$. Here, $\delta =\mathrm{\Delta }-1$ or $\delta =\mathrm{\Delta }-2$ depending on the parity of $n\mathrm{\Delta }$. In this paper, we make progress on the two conjectures. To be precise, we disprove the first conjecture for all $\mathrm{\Delta }\ge 3$ by showing that the limit superior is at most ${\pi }^2\mathrm{/}2$. For small $\mathrm{\Delta }$, we determine the precise asymptotic behavior of $\mathrm{\Delta }-{\lambda }_1(G)$. In particular, we show that $\lim\limits_{n\to\infty} n^2 (\Delta - \lambda_1(G)) /(\Delta - 1) = \pi^2/4$if$\Delta=3$;and$\lim\limits_{n\to\infty} n^2 (\Delta - \lambda_1(G)) /(\Delta - 2) = \pi^2/2$if$\Delta = 4$\mathrm{.}Wealsoconfirmthesecondconjecturefor$\Delta = 3$and$\Delta = 4$ by determining the precise structure of extremal graphs. Particularly, we show that the extremal graphs for $\mathrm{\Delta }\in \{3,4\}$ must have a path-like structure built from specific blocks.