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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 

References

Albert, R. and Barabási, A.-L. (2002). Statistical mechanics of complex networks. Rev. Modern Phys. 47, 47–97.CrossRefGoogle Scholar
Andova, V. et al. (2011). On the Zagreb index inequality of graphs with prescribed vertex degrees. Discrete Appl. Math. 159, 852–858.CrossRefGoogle Scholar
Barabási, A.-L. and Albert, R. (1999). Emergence of scaling in random networks. Science 286, 509–512.CrossRefGoogle ScholarPubMed
Barysz, M., Plavšić, D. and Trinajstić, N. (1986). A note on topological indices. MATCH Commun. Math. Comput. Chem. 19, 89–116.Google Scholar
Bollobás, B. and Riordan, O. (2004). The diameter of a scale-free random graph. Combinatorica 24, 5–34.CrossRefGoogle Scholar
Bollobás, B., Riordan, O., Spencer, J. and Tusnády, G. (2001). The degree sequence of a scale-free random graph process. Random Structures Algorithms 18, 279–290.CrossRefGoogle Scholar
Dondajewski, M. and Szymański, J. (2009). Branches in scale-free trees. J. Math. Sci. 161, 961–968.CrossRefGoogle Scholar
Dorogovtsev, S. N. and Mendes, J. F. F. (2003). Evolution of Networks. Oxford University Press.CrossRefGoogle Scholar
Drmota, M. (2009). Random Trees. Springer Wien New York, Vienna.CrossRefGoogle Scholar
Durrett, R. (2007). Random Graphs Dynamics. Cambrige University Press.Google Scholar
Feng, Q. and Hu, Z. (2011). On the Zagreb index of random recursive trees. J. Appl. Prob. 48, 1189–1196.CrossRefGoogle Scholar
Gordon, M. and Scantlebury, G. R. (1964). Non-random polycondensation: statistical theory of the substitution effect. Trans. Faraday Soc. 60, 604–621.CrossRefGoogle Scholar
Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application. Academic Press, New York.Google Scholar
Janson, S. (2005). Asymptotic degree distribution in random recursive trees. Random Structures Algorithms 26, 69–83.CrossRefGoogle Scholar
Katona, Z. (2005). Width of a scale-free tree. J. Appl. Prob. 42, 839–850.CrossRefGoogle Scholar
Katona, Z. (2006). Levels of a scale-free tree. Random Structures Algorithms 29, 194–207.CrossRefGoogle Scholar
Kim, D.-H., Noh, J. D. and Jeong, H. (2004). Scale-free trees: the skeletons of complex networks. Phys. Rev. E 70, 046126, 5 pp.CrossRefGoogle ScholarPubMed
Li, X., Li, Z. and Wang, L. (2003). The inverse problems for some topological indices in combinatorial chemistry. J. Comput. Biol. 10, 47–55.CrossRefGoogle ScholarPubMed
Mahmoud, H., Smythe, R. and Szymański, J. (1993). On the structure of plane-oriented recursive trees and their branches. Random Structures Algorithms 4, 151–176.CrossRefGoogle Scholar
Móri, T. F. (2002). On random trees. Studia Sci. Math. Hung. 39, 143–155.Google Scholar
Móri, T. F. (2005). The maximum degree of the Barabási–Albert random trees. Combinatorics Prob. Comput. 14, 339–348.CrossRefGoogle Scholar
Neininger, R. (2002). The Wiener index of random trees. Combinatorics Prob. Comput. 11, 587–597.CrossRefGoogle Scholar
Nikiforov, V. (2007). The sum of the squares of degrees: sharp asymptotics. Discrete Math. 307, 3187–3193.CrossRefGoogle Scholar
Nikolić, S., Kovačević, G., Miličević, A. and Trinajstić, N. (2003). The Zagreb indices 30 years after. Croatica Chemica ACTA 76, 113–124.Google Scholar
Peled, U. N., Petreschi, R. and Sterbini, A. (1999). (n, e)-graphs with maximum sum of squares of degrees. J. Graph Theory 31, 283–295.3.0.CO;2-H>CrossRefGoogle Scholar
Platt, J. R. (1947). Influence of neighbor bonds on additive bond properties in paraffins. J. Chem. Phys. 15, 419–420.CrossRefGoogle Scholar