Two vertex colorings of a graph $G$ are equivalent if they induce the same partition of the vertex set into color classes. The graphical Bell number $\mathcal{B}(G)$ is the number of non-equivalent vertex colorings of $G$. We determine a sharp lower bound on $\mathcal{B}(G)$ for graphs $G$ of order $n$ and maximum degree $n-3$, and we characterize the graphs for which the bound is attained.