The Generalised Randić Index of Trees

The Generalised 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 βc = βc(α) such that for all trees on at least 3 vertices R−α(T ) ≤ βc(n + 1) where n = |V (T )| is the number of vertices of T . For example, when α = 1, βc = 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 ) ≥ βc(n − 1). More generally, fix γ > 0 and define ñ = (n−n1)+γn1, where n is the number of vertices of T and n1 is the number of leaves of T . We determine the best constant βc = βc(α, γ) such that for all trees R−α(T ) ≤ βc(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.