| The Randic index of an organic molecule whose molecular graph is G is the sum of the weights (d(u)d(v))-1/2 of all edges uv of G, where d(u) denotes the degree of the vertex u of the molecular graph G.In this thesis we investigate some graphic transformations and properties of extremal graphs on the Randic index among all trees with k pendant vertices, and use some of these transformations to study the problem of extremal values of the Randic index. Among all trees of order n with k pendant vertices, Hansen et al. ([18]) proved that the Randic index of the tree S1,k-1n is minimum, where the tree S1,k-1n is obtained from the star K1,k by attaching the path of length n - k-1 to a pendant vertices of K1,k Li xueliang et al.([26]) showed that the tree S2,k-2n has the second-minimum Randic index, where the tree S2,k-2n is created from the star K1,k by attaching paths of lengths p1,p2(p1, p2 ≥ 1, p1 +P2 = n - k - 1) to two pendant vertices of K1,k, respectively. On this basis, we use some graphic transformations to investigate the extremal values with respect to the Randic index, and characterize the trees with the third-, the fourth-minimum Randic index of order n and k pendant vertices.
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