Maxima of spectral radius of irregular graphs with given maximum degree

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.