COMPUTING ZAGREB INDICES AND
ZAGREB POLYNOMIALS OF FULLERENE,
BUTTERFLY AND BENES NETWORKS
Jianzhong Xu1, Muhammad K. Siddiqui2,
Mohammad R. Farahani3, Ismail Naci Cangul4
1Department of Electronics and Information Engineering
Bozhou University, Bozhou 236800, CHINA
2Department of Mathematics
COMSATS University Islamabad
Lahore Campus, 54000, PAKISTAN
3 Department of Applied Mathematics
Iran University of Science and Technology (IUST)
Narmak, Tehran 16844, IRAN
4Bursa Uludag University, Mathematics
16059, Bursa, TURKEY

Abstract

A topological index is a numerical parameter of a graph which characterizes some of the topological properties of the graph. The concepts of hyper-Zagreb index, first multiple Zagreb index, second multiple Zagreb index and relatedly the Zagreb polynomials were established in chemical graph theory by means of the vertex degrees. It is reported that these indices are useful in the study of anti-inflammatory activities of certain chemical networks. In this paper, we study fullerene, butterfly, Benes networks and determine the hyper-Zagreb index, first multiple Zagreb index, second multiple Zagreb index and the Zagreb polynomials of them.

Citation details of the article



Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 32
Issue: 6
Year: 2019

DOI: 10.12732/ijam.v32i6.1

Download Section



Download the full text of article from here.

You will need Adobe Acrobat reader. For more information and free download of the reader, please follow this link.

References

  1. [1] T. Al-Fozan, P. Manuel, I. Rajasingh, R. S. Rajan, A new technique to compute Padmakar-Ivan index and Szeged index of pericondensed benzenoid graphs, J. Comput. Theor. Nanosci. 11 (2014), 533-539.
  2. [2] A. Alwardi, A. Alqesmah, R. Rangarajan, I. N. Cangul, Entire Zagreb indices of graphs, Discrete Mathematics, Algorithms and Applications, 10, No 3 (2018), Art. 1850037 (16 pages); DOI: 10.1142/S1793830918500374.
  3. [3] M. Bača, A. Shabbir, Total labeling of toroidal polyhexes, Sci. Int.(Lahore) 24, No 3 (2012), 239-241.
  4. [4] V. E. Beneš, Mathematical Theory of Connecting Networks and Telephone Traffic, Academic Press (1965).
  5. [5] A. R. Bindusree, I. N. Cangul, V. Lokesha, A. S. Cevik, Zagreb polynomials of three graph operators, Filomat 30, No 7 (2016), 1979-1986.
  6. [6] K. C. Das, N. Akgunes, M. Togan, A. Yurttas, I. N. Cangul, A. S. Cevik, On the first Zagreb index and multiplicative Zagreb coindices of graphs, Analele Stiint. ale Universitatii Ovidius Constanta, 24, No 1 (2016), 153176.
  7. [7] K. C. Das, A. Yurttas, M. Togan, I. N. Cangul, A. S. Cevik, The multiplicative Zagreb indices of graph operations, J. of Inequalities and Applications, 90 (2013), 1-14; doi:10.1186/1029-242X-2013-90.
  8. [8] M. Deza, P. W. Fowler, A. Rassat, K. M. Rogers, Fullerenes as tilings of surfaces, J. Chem. Inf. Comput. Sci., 40 (2000) 550-558.
  9. [9] A. A. Dobrynin, R. Entringer, I. Gutman, Wiener index of trees: Theory and applications, Acta Appl. Math., 66 (2001) 211-249.
  10. [10] M. Eliasi, A simple approach to order the multiplicative Zagreb indices of connected graphs, Trans. Comb., 4 (2012), 17-24.
  11. [11] M. Eliasi, A. Ghalavand, Ordering of trees by multiplicative second Zagreb index, Trans. Comb., 5, No 1 (2016), 49-55.
  12. [12] M. Eliasi, A. Iranmanesh, I. Gutman, Multiplicative version of first Zagreb index, MATCH Commun. Math. Comput. Chem., 68 (2012), 217-230.
  13. [13] E. Estrada, L. Torres, L. Rodriguez, I. Gutman, An atom-bond connectivity index: modeling the enthalpy of formation of alkanes, Indian J. Chem., 37 (1998), 849-855.
  14. [14] F. Falahati-Nezhad, A. Iranmanesh, A. Tehranian, M. Azari, Comparing the second multiplicative Zagreb coindex with some graph invariants, Trans. Comb., 3, No 4 (2014), 31-41.
  15. [15] B. Furtula, I. Gutman, M. Dehmer, On structure-sensitivity of degreebased topological indices, Appl. Math. Comput., 219 (2013), 8973-8978.
  16. [16] M. Ghorbani, N. Azimi, Note on multiple Zagreb indices, Iran. J. Math. Chem., 3, No 2 (2012), 137-143.
  17. [17] A. Graovac, M. Ghorbani, M. A. Hosseinzadeh, Computing fifth geometric arithmetic index for nanostar dendrimers, J. Math. Nanosci., 1 (2011), 33-42.
  18. [18] I. Gutman, Degree-based topological indices, Croat. Chem. Acta, 86 (2013), 351-361.
  19. [19] I. Gutman, K. C. Das, Some properties of the second Zagreb index, MATCH Commun. Math. Comput. Chem., 50 (2004), 103-112.
  20. [20] I. Gutman, M. Kamran Jamil, N. Akhter, Graphs with fixed number of pendent vertices and minimal first Zagreb index, Trans. Comb., 4, No 1 (2015), 43-48.
  21. [21] I. Gutman, O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer-Verlag, New York (1986).
  22. [22] I. Gutman, N. Trinajstic, Graph theory and molecular orbitals. Total φelectron energy of alternant hydrocarbons, Chem. Phys. Lett., 17 (1972), 535-538.
  23. [23] N. Habibi, T. Dehghan-Zadeh, A. R. Ashrafi, Extremal tetracyclic graphs with respect to the first and second Zagreb indices, Trans. Comb., 5, No 4 (2016), 35-55.
  24. [24] S. Hayat, M. Imran, Computation of certain topological indices of nanotubes covered by C5 and C7 , J. Comput. Theor. Nanosci. 12 (2015), 1-9.
  25. [25] S. Hayat, M. Imran, Computation of topological indices of certain networks, Appl. Math. Comput., 240 (2014), 213-228.
  26. [26] S. A. Hosseini, M. B. Ahmadi, I. Gutman, Kragujevac trees with minimal atom-bond connectivity index, MATCH Commun. Math. Comput. Chem., 71 (2014), 5-20.
  27. [27] S. Imran, M. Hussain, M. K. Siddiqui, M. Numan, Super face d-antimagic labeling for disjoint union of toroidal fullerenes, . Math. Chem., 55, No 3 (2017), 849-863.
  28. [28] R. Kazemi, Probabilistic analysis of the first Zagreb index, Trans. Comb., 2, No 2 (2013), 35-40.
  29. [29] M. Imran, S. Hayat, M. Y. H. Mailk, On topological indices of certain interconnection networks, Appl. Math. Comput. 244 (2014), 936-951.
  30. [30] D. J. Klein, Elemental benzenoids, J. Chem. Inf. Comput. Sci., 34 (1994), 453-459.
  31. [31] S. Konstantinidou, The selective extra stage butterfly, IEEE Trans. Very Large Scale Integr. (VLSI) Syst., 1 (1992), 502-506.
  32. [32] X. Liu, Q. P. Gu, Multicasts on WDM all-optical butterfly networks, J. Inf. Sci. Eng., 18 (2002), 1049-1058.
  33. [33] P. D. Manuel, M.I. Abd-El-Barr, I. Rajasingh, B. Rajan, An efficient representation of Benes networks and its applications, J. Discrete Algorithms, 6 (2008), 11-19.
  34. [34] J. Rada, R. Cruz, I. Gutman, Benzenoid systems with extremal vertexdegree-based topological indices, MATCH Commun. Math. Comput. Chem., 72 (2014), 125-136.
  35. [35] P. S. Ranjini, V. Lokesha, I. N. Cangul, On the Zagreb Indices of the Line Graphs of the Subdivision Graphs, Appl. Math. and Comput., 218, No 3 (2011), 699-702.
  36. [36] G. H. Shirdel, H. RezaPour, A. M. Sayadi, The hyper-Zagreb Index of graph operations, Iran. J. Math. Chem. 4, No 2 (2013), 213-220.
  37. [37] M. K. Siddiqui, M. Imran, A. Ali, On Zagreb indices, Zagreb polynomials of some nanostar dendrimers, Appl. Math. Comput., 280 (2016), 132-139.
  38. [38] M. Togan, A. Yurttas, I. N. Cangul, All versions of Zagreb indices and coindices of subdivision graphs of certain graph types, Advanced Studies in Contemp. Math., 26, No 1 (2016), 227-236.
  39. [39] M. Togan, A. Yurttas, I. N. Cangul, Zagreb and multiplicative Zagreb indices of r-subdivision graphs of double graphs, Scientia Magna, 12, No 1 (2017), 115-119.
  40. [40] H. Wiener, Structural determination of paraffin boiling points, J. Amer. Chem. Soc., 69 (1947), 17-20.
  41. [41] A. William, A. Shanthakumari, Minimum cycle covers of butterfly and Benes network, Int. J. Math. Soft Comput., 2, No 1 (2012), 93-98.
  42. [42] J. Xu, Topological Structure and Analysis of Interconnection Networks, Kluwer Academic Publishers (2001).