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

Abstract: For integers $n\ge3$ and $2\leΔ\le n-1$, let $λ<em>1(n,Δ)$ be the maximum adjacency spectral radius among all connected nonregular graphs of order $n$ and maximum degree $Δ$. Liu conjectured that for each fixed integer $Δ\ge3$, [ \lim{n\to\infty}n^2\bigl(Δ-λ_1(n,Δ)\bigr)= \begin{cases} (Δ-1)π^2/4,&\text{if $Δ$ is odd};\ (Δ-2)π^2/2,&\text{if $Δ$ is even}. \end{cases} ] He proved the case $Δ=3$ and $Δ=4$. We prove the conjecture when $Δ\ge5$ is odd.