The generalized Zagreb index for non-plane and plane recursive trees
Qunqiang Feng, Michael Fuchs, Tsan-Cheng Yu
TL;DR
This work examines the generalized Zagreb index $Z_n^{(k)}$ on two random recursive tree models: non-plane recursive trees and plane recursive trees. It develops the moment-transfer method to analyze distributional recurrences for $Z_n^{(k)}$ and the root degree $R_n$, deriving first-order moment asymptotics and limit laws. For non-plane recursive trees, $(Z_n^{(k)}-\mu_k n)/(\sigma_k \sqrt{n})$ converges to a normal distribution for all $k\ge 2$, while for plane recursive trees the index exhibits non-normal limits for $k=3$ and $4$ with explicit moment recurrences; the $k=2$ case aligns with known results and highlights the method’s consistency. The paper provides explicit mean and higher-moment asymptotics, introduces a robust framework where martingale or Stein methods are difficult to apply, and advances understanding of Zagreb-type indices in random trees with potential implications for chemical graph theory and network analysis.
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.