The generalized Zagreb index for non-plane and plane recursive trees

Abstract

The Zagreb index, which is defined as the sum of squares of degrees of the nodes of a tree, was studied in previous works by martingale techniques for random non-plane recursive trees and classes of random trees which are close to random plane recursive trees. These techniques are not easily amended to the generalized Zagreb index, which is defined similar but with squares replaced by higher powers. In this paper, we use the moment transfer approach to (i) obtain the first-order asymptotics of moments and to (ii) prove limit laws for the (suitable normalized) generalized Zagreb index for random non-plane and plane recursive trees; for the former, we show that for all higher powers the limit law is normal, for the latter, we show for cubes and fourth powers that its a non-normal law.