The generalized randic index of trees

The generalized Randic index R-alpha(T) of a tree T is the sum over the edges uv of T of (d(u)d(v))(-alpha) where d(x) is the degree of the vertex x in T For all alpha > 0, we find the minimal constant beta(0) = beta(0)(a) such that for all trees on at least 3 vertices, R-alpha(T) <= beta(0)(n + 1), where n = n(T) = vertical bar V(T)vertical bar is the number of vertices of T. For example, when a = 1, beta(0) = 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-alpha(T) >= beta(0)(n - 1). More generally, fix gamma > 0 and define i = (n - n(1)) + gamma n(1), where n(1) = n(1)(T) is the number of leaves of T. We determine the best constant beta(0) = beta(0)(alpha, gamma) such that for all trees on at least 3 vertices, R-alpha(T) < beta(0)(n + 1). Using these results one can determine (up to O(n) terms) the maximal Randic 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. (c) 2007 Wiley Periodicals, Inc.