Trees with small Randic connectivity indices

Let T be a tree and d(v) the degree of its vertex v. Then the connectivity index, or Randic index, of T is defined as chi(T) = Sigma(uv) 1/rootd(u)d(v), where the summation goes over all edges uv of T. In the existing literature, trees of order n with m pending vertices and with the smallest connectivity index were determined by Hansen et al, whereas the unique tree of order n with the smallest connectivity index was determined by Bollobas et al. In this paper, we determine all trees of order n with m pending vertices and with the second smallest connectivity index and all trees of order n with diameter r and with the smallest and the second smallest connectivity indices. The unique tree of order n with, respectively, the second, the third and the fourth smallest connectivity index is also determined.