Bounds For The Randic Index R_-1

The Randic index R_-1 = R_-1(G) of a graph G is defined as the sum of the weights (d_G(u)d_G(v))^-1 of all edges uv of G, where d_G(u) denotes the degree of vertex u. In section 2 of this paper we give a sharp lower bound for the Randic index R_-1 of trees with given order and the number of pending vertices, and determine the trees of order n ≥ 6 with second-minimal, third-minimal and fourth-minimal Randic indices R_-1. In section 3, we give a sharp lower bound for the Randic index R_-1 of starlike trees with given order, determine the starlike tree with second-minimal and third-minimal Randic indices R_-1 and prove that the Randic index R_-1 is monotone increasing over the well order sequence of starlike trees.