Skip to main content
SpringerLink
Log in

On topological properties of hierarchical interconnection networks

  • Original Research
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

There is really a lot of mathematics involved in electrical and electronic engineering. It depends on what area of electrical and electronic engineering, for example there is a lot more abstract mathematics in communication theory and signal processing and networking etc. Networks involve nodes communicating with each other. A lot of computers linked together form a network. Cell phone users form a network. Networking involves the study of the best way of implementing a network. Graph theory has found a considerable use in this area of research. In this paper, we extend this study to interconnection networks and derive analytical closed results of general Randić index \(R_{\alpha }(G)\) for different values of “\(\alpha \)” for block shift network (BSN-1) and (BSN-2), Hierarchical hypercube (HHC-1) and (HHC-2). We also compute first Zagreb, \(\textit{ABC}\), \(\textit{GA}\), \(\textit{ABC}_{4}\) and \(\textit{GA}_{5}\) indices and give closed formulae of these indices for Hierarchical interconnection networks.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Abd-El-Barr, M., Al-Somani, T.F.: Topological properties of hierarchical interconnection networks: a review and comparison. J. Electr. Comput. Eng. (2011). doi:10.1155/2011/189434

  2. Bača, M., Horváthová, J., Mokrišová, M., Suhányiovǎ, A.: On topological indices of fullerenes. Appl. Math. Comput. 251, 154–161 (2015)

    MathSciNet  MATH  Google Scholar 

  3. Baig, A.Q., Imran, M., Ali, H.: Computing Omega, Sadhana and PI polynomials of benzoid carbon nanotubes. Optoelectron. Adv. Mater. Rapid Commun. 9, 248–255 (2015)

    Google Scholar 

  4. Baig, A.Q., Imran, M., Ali, H.: On topological indices of poly oxide, poly silicate, DOX and DSL networks. Can. J. Chem. 93, 730739 (2015)

    Article  Google Scholar 

  5. Diudea, M.V., Gutman, I., Lorentz, J.: Molecular Topology. Nova, Huntington (2001)

    Google Scholar 

  6. Estrada, E., Torres, L., Rodríguez, L., Gutman, I.: An atom–bond connectivity index: modelling the enthalpy of formation of alkanes. Indian J. Chem. 37A, 849–855 (1998)

    Google Scholar 

  7. Ghorbani, M., Hosseinzadeh, M.A.: Computing \(ABC_{4}\) index of nanostar dendrimers. Optoelectron. Adv. Mater. Rapid Commun. 4, 1419–1422 (2010)

    Google Scholar 

  8. Graovac, A., Ghorbani, M., Hosseinzadeh, M.A.: Computing fifth geometric–arithmetic index for nanostar dendrimers. J. Math. Nanosci. 1, 3342 (2011)

    Google Scholar 

  9. Gutman, I., Polansky, O.E.: Mathematical Concepts in Organic Chemistry. Springer, New York (1986)

    Book  MATH  Google Scholar 

  10. Hayat, S., Imran, M.: Computation of topological indices of certain networks. Appl. Math. Comput. 240, 213–228 (2014)

    MathSciNet  MATH  Google Scholar 

  11. Hayat, S., Imran, M.: Computation of certain topological indices of nanotubes. J. Comput. Theor. Nanosci. 12, 70–76 (2015)

    Article  Google Scholar 

  12. Hayat, S., Imran, M.: Computation of certain topological indices of nanotubes covered by \(C_{5}\) and \(C_{7}\). J. Comput. Theor. Nanosci. 12, 533–541 (2015)

    Article  Google Scholar 

  13. Hayat, S., Imran, M.: On some degree based topological indices of certain nanotubes. J. Comput. Theor. Nanosci. 12, 1599–1605 (2015)

    Article  Google Scholar 

  14. Huiqing, L.I.U., Zheng, Y.A.N., Heguo, L.I.U.: Extremal chemical (n, m, k)-graphs with maximum Randić index. MATCH Commun. Math. Comput. Chem. 60, 513–522 (2008)

    MathSciNet  MATH  Google Scholar 

  15. Iranmanesh, A., Zeraatkar, M.: Computing GA index for some nanotubes. Optoelectron. Adv. Mater. Rapid Commun. 4, 1852–1855 (2010)

    Google Scholar 

  16. Konstantinidou, S.: The selective extra stage butterfly. IEEE Trans. Very Large Scale Integr. (VLSI) Syst. 1, 502–506 (1992)

    Google Scholar 

  17. Manuel, P.D., Abd-El-Barr, M.I., Rajasingh, I., Rajan, B.: An efficient representation of Benes networks and its applications. J. Discret. Algorithms 6, 11–19 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Othmer, H.G.: A Graph-Theoretic Analysis of Chemical Reaction Networks. University of Utah, Salt Lake City (1981)

    Google Scholar 

  19. Randić, M.: On characterization of molecular branching. J. Am. Chem. Soc. 97, 6609–6615 (1975)

    Article  Google Scholar 

  20. Vukičević, D., Furtula, B.: Topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges. J. Math. Chem. 46, 1369–1376 (2009)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Government College University, Faisalabad, Pakistan

    Haidar Ali & Muhammad Kashif Shafiq

  2. Department of Mathematics, COMSATS Institute of Information Technology, Attock, Pakistan

    A. Q. Baig

Authors
  1. Haidar Ali
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Q. Baig
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Muhammad Kashif Shafiq
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Haidar Ali.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ali, H., Baig, A.Q. & Shafiq, M.K. On topological properties of hierarchical interconnection networks. J. Appl. Math. Comput. 55, 313–334 (2017). https://doi.org/10.1007/s12190-016-1038-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-016-1038-3

Keywords

Mathematics Subject Classification

Search

Navigation