Journal of Applied Probability

Phase changes in the topological indices of scale-free trees

Qunqiang Feng and Zhishui Hu

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

First available in Project Euclid: 19 June 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Random network small world Zagreb index scale-free tree


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.

Export citation


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