Eccentricity and algebraic connectivity of graphs

Let G be a graph on n nodes with algebraic connectivity ₂. The eccentricity of a node is defined as the length of a longest shortest path starting at that node. If s_ denotes the number of nodes of eccentricity at most, then for 2, ₂ 4 \, s_ (-2+4{n) \, n² }. As a corollary, if d denotes the diameter of G, then ₂ 4 (d-2+4{n) \, n }. It is also shown that ₂ s_ 1+ (e (G^{) -m) }, where m and e (G^) denote the number of edges in G and in the -th power of G, respectively.