A scale-free tree with the parameter β is very close to a star if β is just a bit larger than -1, whereas it is close to a random recursive tree if β is very large. Through the Zagreb index, we consider the whole scene of the evolution of the scale-free trees model as β goes from -1 to + ∞. The critical values of β are shown to be the first several nonnegative integer numbers. We get the first two moments and the asymptotic behaviors of this index of a scale-free tree for all β. The generalized plane-oriented recursive trees model is also mentioned in passing, as well as the Gordon-Scantlebury and the Platt indices, which are closely related to the Zagreb index.
"Phase changes in the topological indices of scale-free trees." J. Appl. Probab. 50 (2) 516 - 532, June 2013. https://doi.org/10.1239/jap/1371648958