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A Measure for Quantifying the Topological Structure of Some Networks

  • Meryam ZeryouhEmail author
  • Mohamed El Marraki
  • Mohamed Essalih
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11028)

Abstract

Determining and quantifying the topological structure of networks is an exciting research topic in theoretical network science. For this purpose, a large amount of topological indices have been studied. They function as effective measures for improving the performance of existing networks and designing new robust networks. In this paper, we focus on a distance-based graph invariant named the Terminal Wiener index. We use this measure to analyze the structure of two well-known hierarchical networks: the Dendrimer tree \(\mathcal{T}_{d,h}\) and the Dendrimer graph \(\mathcal{D}_{d,h}\). We also investigate two methods of calculation in order to show that the proposed method reduces the computational complexity of the Terminal Wiener index.

Keywords

Networks Topological indices Terminal Wiener index Dendrimer tree Dendrimer graph Computational complexity 

References

  1. 1.
    Emmert-Streib, F., Dehmer, M.: Networks for systems biology: conceptual connection of data and function. IET Syst. Biol. 5(3), 185–207 (2011)CrossRefGoogle Scholar
  2. 2.
    Wiener, H.: Structural determination of paraffin boiling points. J. Am. Chem. Soc. 69, 17–20 (1947)CrossRefGoogle Scholar
  3. 3.
    Wuchty, S., Stadler, P.F.: Centers of complex networks. J. Theor. Biol. 223(1), 45–53 (2003)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Estrada, E., Vargas-Estrada, E.: Distance-sum heterogeneity in graphs and complex networks. Appl. Math. Comput. 218(21), 10393–10405 (2012)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Kraus, V., Dehmer, M., Emmert-Streib, F.: Probabilistic inequalities for evaluating structural network measures. Inf. Sci. 288, 220–245 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Gutman, I., Furtula, B., Petrovic, M.: Terminal Wiener index. J. Math. Chem. 46, 522–531 (2009)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Rodríguez-Velázquez, J.A., Kamis̃alić, A., Domingo-Ferrer, J.: On reliability indices of communication networks. Comput. Math. Appl. 58(7), 1433–1440 (2009)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Goel, S., Anderson, A., Hofman, J., Watts, D.J.: The structural virality of online diffusion. Manag. Sci. 62(1), 180–196 (2015)Google Scholar
  9. 9.
    Mohar, B., Pisanski, T.: How to compute the Wiener index of graph. J. Math. Chem. 2(3), 267–277 (1988)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Klavz̃ar, S.: On the canonical metric representation, average distance, and partial Hamming graphs. Eur. J. Comb. 27(1), 68–73 (2006)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Klajnert, B., Bryszewska, M.: Dendrimers: properties and applications (2001)Google Scholar
  12. 12.
    Essalih, M., El Marraki, M., Alhagri, G.: Calculation of some topological indices graph. J. Theor. Appl. Inf. Technol. 30(2), 122–128 (2011)Google Scholar
  13. 13.
    Klavz̃ar, S., Gutman, I.: Wiener number of vertex-weighted graphs and a chemical application. Discret. Appl. Math. 80(1), 73–81 (1997)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Graham, R.L., Winkler, P.M.: On isometric embeddings of graphs. Trans. Am. Math. Soc. 288(2), 527–536 (1985)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Chepoi, V., Klavz̃ar, S.: The Wiener index and the Szeged index of benzenoid systems in linear time. J. Chem. Inf. Comput. Sci. 37(4), 752–755 (1997)CrossRefGoogle Scholar
  16. 16.
    C̃repnjak, M., Tratnik, N.: The Szeged index and the Wiener index of partial cubes with applications to chemical graphs. Appl. Math. Comput. 309, 324–333 (2017)MathSciNetGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Meryam Zeryouh
    • 1
    Email author
  • Mohamed El Marraki
    • 1
  • Mohamed Essalih
    • 2
  1. 1.LRIT - CNRST URAC 29, Rabat IT Center - Faculty of SciencesMohammed V University in RabatRabatMorocco
  2. 2.LAPSSII, The Safi’s Graduate Shcool of TechnologyCadi Ayyad University in MarrakeshMarrakeshMorocco

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