Extremal spectral radius of nonregular graphs with a fixed odd maximum degree

For integers $n\ge 3$ and $2\le \mathrm{\Delta }\le n-1$, let ${\lambda }_1(n,\mathrm{\Delta })$ be the maximum adjacency spectral radius among all connected nonregular graphs of order $n$ and maximum degree $\mathrm{\Delta }$. Liu conjectured that for each fixed integer $\mathrm{\Delta }\ge 3$, $$\underset{n\to \mathrm{\infty }}{\lim }{n}^2(\mathrm{\Delta }-{\lambda }_1(n,\mathrm{\Delta }))=\{\begin{array}{ll}(\mathrm{\Delta }-1){\pi }^2\mathrm{/}4,& \text{if }\mathrm{\Delta }\text{ is odd};\\ (\mathrm{\Delta }-2){\pi }^2\mathrm{/}2,& \text{if }\mathrm{\Delta }\text{ is even}\mathrm{.}\end{array}$$ He proved the case $\mathrm{\Delta }=3$ and $\mathrm{\Delta }=4$. We prove the conjecture when $\mathrm{\Delta }\ge 5$ is odd.