Maximum tree and maximum value for the randić index R-1 of trees of order n ≤ 102

The Randić index R_-1(G) of a graph G is defined as the sum of the weights (d(u)d(v))^-1 of all edges uv of G, where d(u) denotes the degree of a vertex u in G. Trees with maximum Randić index R_-1 need not be unique. Clark et al. gave the maximum values for the index of trees of order n ≤ 20. In this paper, we determine the maximum value for the Randić index R_-1 of all trees of order n ≤ 102, and give one of the trees with maximum value of the index. This not only largely extends the known range of the orders n of trees with maximum index, but also gives a convincible solution for the induction initial of our previous paper. Because there is a huge number of trees of order n ≤ 102, it is not possible to directly search the trees with maximum index by a computer. Our method is to first figure out the simple structure of one of the trees of order n with maximum R_-1 for each n ≤ 102, i.e., the branching subtree must be a star. Then from this simple structure, we can employ mathematical programming to easily calculate the maximum value of R_-1 for each n.