The generalized Randić index of trees

The generalized Randić; index R-α(T) of a tree T is the sum over the edges uv of T of (d(u)d(v))-α where d(x) is the degree of the vertex x in T. For all α > 0 , we find the minimal constant β₀=β₀(α) such that for all trees on at least 3 vertices, R-α(T)≤β₀(n+1) , where n=n(T)= |V(T)| is the number of vertices of T. For example, when α=1, β₀=15 56 . This bound is sharp up to the additive constant—for infinitely many n we give examples of trees T on n vertices with R-α(T)≥ β₀(n- 1) . More generally, fix γ > 0 and define ñ=(n- n₁)+γ n₁ , where n₁= n₁(T) is the number of leaves of T. We determine the best constant β₀=β₀(α, γ) such that for all trees on at least 3 vertices, R-α(T)≤ β₀(ñ+1) . Using these results one can determine (up to o(n) terms) the maximal Randić; index of a tree with a specified number of vertices and leaves. Our methods also yield bounds when the maximum degree of the tree is restricted.