  Let $\lambda^{*}$ be the maximum spectral radius of connected irregular
graphs on $n$ vertices with maximum degree $\Delta$. Liu, Shen and Wang (2007)
conjectured that $\lim_{n\rightarrow
\infty}(n^{2}(\Delta-\lambda^{*}))/(\Delta-1)=\pi^{2},$ which describes the
asymptotic behavior for the maximum spectral radius of irregular graphs.
Focusing on this conjecture, we consider the maximum spectral radius of
connected subcubic bipartite graphs. The unique connected subcubic bipartite
graph with the maximum spectral radius is determined. Let $G$ be a
$k$-connected irregular graph with spectral radius $\lambda_{1}(G)$, we present
a lower bound for $\Delta-\lambda_{1}(G)$. Moreover, if $H$ is a proper
subgraph of a $k$-connected $\Delta$-regular graph, a lower bound for
$\Delta-\lambda_{1}(H)$ is also obtained. These bounds improve some previous
results.
