Computation of Zagreb indices and Zagreb polynomials of Sierpinski graphs

Year 2020, , 754 - 765, 02.04.2020

Hafiz Muhammad Afzal Siddiqui

https://doi.org/10.15672/hujms.623990

Cited By: 15

Abstract

The Sierpinski fractal or Sierpinski gasket and generalized Sierpinski graphs are objects of great interest in dynamical systems and probability. In this paper, we consider the Sierpinski gasket graph $S_{n}$, the generalized Sierpinski graphs $S(n,C_{3})$ and $S(n,C_{4})$. We provide explicit computing formulae for Zagreb indices, multiple Zagreb indices and Zagreb polynomials of Sierpinski graphs.

Keywords

Sierpinski gasket graph, generalized Sierpinski graph, topological indices, Zagreb index, augmented Zagreb index, Zagreb polynomials

References

• [1] H. Ali, A.Q. Baig and M.K. Shafiq, On topological properties of hierarchical interconnection networks, J. Appl. Math. Comput. 55 (1-2), 313–334, doi:10.1007/s12190- 016-1038-3, 2016.
• [2] A.T. Balaban, I. Motoc, D. Bonchev and O. Makenyan, Topological indices for structure-activity correlations, Topics Curr. Chem. 114, 21–55, 1983.
• [3] G. Caporossi, P. Hansen and D. Vukičević, Comparing of Zagreb indices of cylic graphs, MATCH Commun. Math. Comput. Chem. 63, 441–451, 2010.
• [4] K.C. Das and I. Gutman, Some properties of the second Zagreb index, MATCH Commun. Math. Comput. Chem. 52, 103–112, 2004.
• [5] M.V. Diudea, (Ed.), QSPR/QSAR Studies by molecular descriptors, NOVA, New York, 2001.
• [6] M.R. Farahani, H.M.A. Siddiqui, Sh. Baby, M. Imran and M.K. Siddiqui, The Second and Second Sum connectivity Indices of $TUC_{4}C_{8}$ Nanutubes, J. Optoelectron. Bio. Mater. , 8 (3), 107–111, 2016.
• [7] H. Fath-tabar, Zagreb polynomials and PI indices of some nanostructures, Digest. J. Nanomater. Bios. 4, 189–191, 2009.
• [8] B. Furtula and I. Gutman, A forgotten topological index, J. Math. Chem. 53, 1184- 1190, 2015.
• [9] B. Furtula, A. Graovac and D. Vukičević, Augmented Zagreb index, J. Math. Chem. 48, 370–380, 2010.
• [10] B. Furtula, I. Gutman and M. Dehmer, On structural-sensitivity of degree-based topological indices, Appl.Math. Comput. 219, 8973–8978, 2013.
• [11] M. Ghorbani and N. Azimi, Note on multiple Zagre indices, Iran. J. Math. Chem. 3, 137–143, 2012.
• [12] I. Gutman, Degree-based topological indices, Croat. Chem. Acta, 86, 351–361, 2013.
• [13] I. Gutman, An exceptional property of first Zagreb index, MATCH Commun. Math. Comput. Chem. 72, 733–740, 2014.
• [14] I. Gutman and K.C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50, 83–92, 2004.
• [15] I. Gutman and B. Furtula, Ž. K. Vukićević and G. Popivoda, On Zagreb indices and coindices, MATCH Commun. Math. Comput. Chem. 74, 5–16, 2015.
• [16] I. Gutman and O. Polansky, Mathematical Concepts in Organic Chemistry, Springer- Verlag, Berlin, 1986.
• [17] I. Gutman and J. Tošović, Testing the quality of molecular structures descriptors. Vertex-degree-based topological indices, J. Serb. Chem. Soc. 78, 805–810, 2013.
• [18] I. Gutman and N. Trinajstić, Graph theory and molecular orbitals. Total π-electron energy of alternate hydrocarbons, Chem. Phy. Lett. 17, 535–538, 1972.
• [19] I. Gutman, B. Ruščić, N. Trinajstić and C.F. wilcox, Graph theory and molecular orbitals. XII. Acyclic polyenes, J. Chem. Phys. 62, 1692–1704, 1975.
• [20] S. Hayat and H.M.A. Siddiqui, On bipartite edge frustration of carbon and boron nanotubes, Studia UBB Chemia, LXI(1), 283–290, 2016.
• [21] A.M. Hinz and D. Parisse, Coloring Hanoi and Sierpiński graphs, Disc. Math. 312 (9), 1521–1535, 2012.
• [22] M. Imran, A.Q. Baig, H. Ali and S.U. Rehman, On topological properties of poly honeycomb networks, Period. Math. Hungr. 73 (1), 100–119, 2016.
• [23] A. Jonsson, A trace theorem for the Drichlet form on the Sierpiński gasket, Math. Z. 250, 599–609, 2005.
• [24] H. Narumi and H. Katayama, Simple topological index, a newly devised index characterizing the topological nature of structural isomers of saturated hydrocarbons, Mem. Fac. Engin. Hokkaido Univ. 16, 209–214, 1984.
• [25] S. Nikolić, G. Kovačević, A. Milič and N. Trinajstić, The Zagreb indices 30 years after, Croat. Chem. Acta, 76, 113–124, 2003.
• [26] R.S. Scorer, P.M. Grundy and C.A.B. Smith, Some binary games, Math. Gaz. 28, 96–103, 1944.
• [27] G.H. Shirdel, H. Rezapour and A.M. Sayadi, The hyper-Zagreb index of graph operations, Iran. J. Math. Chem. 4, 213–220, 213.
• [28] S. Wang, M.R. Farahani and A.Q. Baig, The Sadhana polynomial and Sadhana index of polycyclic aromatic hydrocarbon PAHk, J. Chem. Pharm. Res. 8 (6), 526–531, 2016.
• [29] K. Xu and K.Ch. Das, Zagreb indices and polynomials of $TUHRC_{4}$ and $TUSC_{4}C_{8}$ nanotubes, MATCH Commun. Math. Comput. Chem. 68, 257–272, 2012.
• [30] L. Yang, X. Ai and L. Zhang, The Zagreb coindices of a type of composite graph, Hacettepe J. Math. Stat. 45 (4), 1135–1145, 2016.
• [31] B. Zhou, Zagreb indices, MATCH Commun. Math. Comput. Chem. 52, 113–118, 2004.