Journal of Applied Probability

On the Zagreb index of random recursive trees

Qunqiang Feng and Zhishui Hu

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Abstract

We investigate the Zagreb index, one of the topological indices, of random recursive trees in this paper. Through a recurrence equation, the first two moments of Zn, the Zagreb index of a random recursive tree of size n, are obtained. We also show that the random process {Zn - E[Zn], n ≥ 1} is a martingale. Then the asymptotic normality of the Zagreb index of a random recursive tree is given by an application of the martingale central limit theorem. Finally, two other topological indices are also discussed in passing.

Article information

Source
J. Appl. Probab., Volume 48, Number 4 (2011), 1189-1196.

Dates
First available in Project Euclid: 16 December 2011

Permanent link to this document
https://projecteuclid.org/euclid.jap/1324046027

Digital Object Identifier
doi:10.1239/jap/1324046027

Mathematical Reviews number (MathSciNet)
MR2896676

Zentralblatt MATH identifier
1234.05053

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

Keywords
Random tree Zagreb index recursive tree martingale central limit theorem

Citation

Feng, Qunqiang; Hu, Zhishui. On the Zagreb index of random recursive trees. J. Appl. Probab. 48 (2011), no. 4, 1189--1196. doi:10.1239/jap/1324046027. https://projecteuclid.org/euclid.jap/1324046027


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References

  • Ali Khan, T. and Neininger, R. (2007). Tail bounds for the Wiener index of random trees. In 2007 Conference on Analysis of Algorithms, AofA 07 (Discrete Math. Theoret. Comput. Sci. Proc. AH), Association of Discrete Mathematics and Theoretical Computer Science, Nancy, pp. 279–289.
  • Barysz, M., Plavšić, D. and Trinajstić, N. (1986). A note on topological indices. MATCH Commun. Math. Comput. Chem. 19, 89–116.
  • Clark, L. H. and Moon, J. W. (2000). On the general Randić index for certain families of trees. Ars Combinatoria 54, 223–235.
  • Devillersand. J. and Balaban, A. T. (1999). Topological Indices and Related Descriptors in QSAR and QSPR. Gordon and Breach, Amsterdam.
  • Devroye, L. and Lu, J. (1995). The strong convergence of maximal degrees in uniform random recursive trees and dags. Random Structures Algorithms 7, 1–14.
  • Feng, Q. (2011). The Zagreb index of random binary trees. Unpublished manuscript.
  • Feng, Q., Mahmoud, H. M. and Panholzer, A. (2008). Limit laws for the Randić index of random binary tree models. Ann. Inst. Statist. Math. 60, 319–343.
  • Goh, W. and Schmutz, E. (2002). Limit distribution for the maximum degree of a random recursive tree. J. Comput. Appl. Math. 142, 61–82.
  • Gordon, M. and Scantlebury, G. R. (1964). Non-random polycondensation: statistical theory of the substitution effect. Trans. Faraday Soc. 60, 604–621.
  • Gutman, I. and Trinajstić, N. (1972). Graph theory and molecular orbitals. Total $\varphi$-electron energy of alternant hydrocarbons. Chem. Phys. Lett. 17, 535–538.
  • Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application. Academic Press, New York.
  • Harary, F. (1972). Graph Theory. Addison-Wesley, Reading, MA.
  • Hollas, B. (2005). Asymptotically independent topological indices on random trees. J. Math. Chem. 38, 379–387.
  • Janson, S. (2003). The Wiener index of simply generated random trees. Random Structures Algorithms 22, 337–358.
  • Janson, S. and Chassaing, P. (2004). The center of mass of the ISE and the Wiener index of trees. Electron. Commun. Prob. 9, 178–187.
  • Li, X., Li, Z. and Wang, L. (2003). The inverse problems for some topological indices in combinatorial chemistry. J. Comput. Biol. 10, 47–55.
  • Neininger, R. (2002). The Wiener index of random trees. Combinatorics Prob. Comput. 11, 587–597.
  • Nikolić, S., Kovačević, G., Miličević, A. and Trinajstić, N. (2003a). The Zagreb indices 30 years after. Croatica Chemica Acta 76, 113–124.
  • Nikolić, S., Tolić, I. M., Trinajstić, N. and Baučić, I. (2000). On the Zagreb indices as complexity indices. Croatica Chemica Acta 73, 909–921.
  • Nikolić, S. \et (2003b). On molecular complexity indices. In Complexity in Chemistry: Introduction and Fundamentals, eds D. Bonchev and D. H. Rouvray, Taylor and Francis, London, pp. 29–89.
  • Platt, J. R. (1947). Influence of neighbor bonds on additive bond properties in paraffins. J. Chem. Phys. 15, 419–420.
  • Smythe, R. T. and Mahmoud, H. M. (1994). A survey of recursive trees. Teor. Imovir. Mat. Stat. 51, 1–29 (in Ukrainian). English translation: Theory Prob. Math. Statist. 51 (1996), 1–27.
  • Trinajstić, N. (1992). Chemical Graph Theory, 2nd edn. CRC Press, Boca Raton, FL.