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Phase Changes in the Topological Indices of Scale-Free Trees

Part of: Limit theorems Graph theory Combinatorial probability

Published online by Cambridge University Press:  30 January 2018

Qunqiang Feng  and
Zhishui Hu
Qunqiang Feng*
Affiliation:
University of Science and Technology of China
Zhishui Hu*
Affiliation:
University of Science and Technology of China
*
∗ Postal address: Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei, Anhui 230026, China.
∗ Postal address: Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei, Anhui 230026, China.
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Abstract

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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.

Keywords

Random networksmall worldZagreb indexscale-free tree

MSC classification

Primary: 05C05: Trees 60C05: Combinatorial probability
Secondary: 60F05: Central limit and other weak theorems
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 2 , June 2013 , pp. 516 - 532
Copyright
© Applied Probability Trust 

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