On the spectral radius of graphs: nonregular distance-hereditary graphs with given edge-connectivity, graphs with tree-width $k$ and block graphs with prescribed independence number $\alpha$

The edge-connectivity of a graph is the minimum number of edges whose deletion disconnects the graph. Let $\mathrm{\Delta }(G)$ 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 $\mathrm{\Delta }(G)-\rho (G)$ 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.