The generalized randić index of trees
The generalized Randić index R-α(T) of a tree Tis 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 Ton n vertices with R-α(T) ≥ β0(n - 1). More generally, fix γ > 0 and define ñ = (n - n1 + γn1, 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. ©2007 wiley Periodicals, Inc.
Journal of Graph Theory
Balister, P., Bollobás, B., & Gerke, S. (2007). The generalized randić index of trees. Journal of Graph Theory, 56 (4), 270-286. https://doi.org/10.1002/jgt.20267