Eccentricity and algebraic connectivity of graphs

Let $G$ be a graph on $n$ nodes with algebraic connectivity ${\lambda }_2$. The eccentricity of a node is defined as the length of a longest shortest path starting at that node. If $s_{\mathrm{\ell }}$ denotes the number of nodes of eccentricity at most $\mathrm{\ell }$, then for $\mathrm{\ell }\ge 2$, $${\lambda }_2\ge \frac{4s_{\mathrm{\ell }}}{(\mathrm{\ell }-2+\frac{4}{n})n^2}\mathrm{.}$$ As a corollary, if $d$ denotes the diameter of $G$, then $${\lambda }_2\ge \frac{4}{(d-2+\frac{4}{n})n}\mathrm{.}$$ It is also shown that $${\lambda }_2\ge \frac{s_{\mathrm{\ell }}}{1+\mathrm{\ell }(e(G^{\mathrm{\ell }})-m)},$$ where $m$ and $e(G^{\mathrm{\ell }})$ denote the number of edges in $G$ and in the $\mathrm{\ell }$-th power of $G$, respectively.