  A graph is trivial if it contains one vertex and no edges. The essential
connectivity $\kappa^{\prime}$ of $G$ is defined to be the minimum number of
vertices of $G$ whose removal produces a disconnected graph with at least two
non-trivial components. Let $\mathcal{A}_n^{\kappa',\delta}$ be the set of
graphs of order $n$ with minimum degree $\delta$ and essential connectivity
$\kappa'$. In this paper, we determine the graphs attaining the maximum
spectral radii among all graphs in $\mathcal{A}_n^{\kappa',\delta}$ and
characterize the corresponding extremal graphs. In addition, we also determine
the digraphs which achieve the maximum spectral radii among all strongly
connected digraphs with given essential connectivity and give the exact values
of the spectral radii of these digraphs.
