  The edge-connectivity of a graph is the minimum number of edges whose
deletion disconnects the graph. Let $\Delta(G)$ the maximum degree of a graph
$G$ and let $\rho(G)$ be the spectral radius of $G$. In this article we present
a lower bound for $\Delta(G)-\rho(G)$ in terms of the edge connectivity of $G$,
where $G$ is a nonregular distance-hereditary graph. We also prove that
$\rho(G)$ reaches the maximum at a unique graph in $\mathcal G$, when $\vert
V(G)\vert = n$, and $\mathcal G$ either is in the class of graphs with bounded
tree-width or is in the class of block graphs with prescribed independence
number.
