## Journal of Applied Probability

### Phase changes in the topological indices of scale-free trees

#### Abstract

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.

#### Article information

Source
J. Appl. Probab., Volume 50, Number 2 (2013), 516-532.

Dates
First available in Project Euclid: 19 June 2013

https://projecteuclid.org/euclid.jap/1371648958

Digital Object Identifier
doi:10.1239/jap/1371648958

Mathematical Reviews number (MathSciNet)
MR3102497

Zentralblatt MATH identifier
1267.05065

Subjects
Primary: 05C05: Trees 60C05: Combinatorial probability
Secondary: 60F05: Central limit and other weak theorems

#### Citation

Feng, Qunqiang; Hu, Zhishui. Phase changes in the topological indices of scale-free trees. J. Appl. Probab. 50 (2013), no. 2, 516--532. doi:10.1239/jap/1371648958. https://projecteuclid.org/euclid.jap/1371648958

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