Let $G$ be a connected nonregular graphs of order $n$ with maximum degree $Δ$ that attains the maximum spectral radius. Liu and Li (2008) proposed a conjecture stating that $G$ has a degree sequence $(Δ,\ldots,Δ,δ)$ with $δ<Δ$. For $Δ=3$ and $Δ=4$, Liu (2024) confirmed this conjecture by characterizing the structure of such graphs. Liu also proposed a modified version of the conjecture for fixed $Δ$ and sufficiently large $n$, stating that the above $δ=Δ-1$ if $Δ$ and $n$ are both odd, $δ=1$ if $Δ$ is odd and $n$ is even, and $δ=Δ-2$ if $Δ$ is even. For the cases where $Δ=n-2$ with $n\ge 5$, and $Δ=n-3$ with $n\ge 59$, we fully characterize the structure of $G$.