Advertisement

Journal of Combinatorial Optimization

, Volume 35, Issue 3, pp 814–841 | Cite as

The Wiener index of Sierpiński-like graphs

  • Chunmei Luo
  • Liancui ZuoEmail author
  • Philip B. Zhang
Article

Abstract

Sierpiński-like graphs constitute an extensively studied family of graphs of fractal nature applicable in topology, mathematics of the Tower of Hanoi, computer science, and elsewhere. In this paper, we focus on the Wiener polarity index, Wiener index and Harary index of Sierpiński-like graphs. By Sierpiński-like graphs’ special structure and correlation, their Wiener polarity index and some Sierpiński-like graph’s Wiener index and Harary index are obtained.

Keywords

Sierpiński-like graphs Wiener polarity index Wiener index Harary index 

References

  1. Bollobás B (1998) Modern graph theory. Springer, New YorkCrossRefzbMATHGoogle Scholar
  2. Beaudou L, Gravier S, Klavžar S, Kovše M, Mollard M (2010) Covering codes in Sierpiński graphs. Discrete Math Theor Comput Sci 12:63–74MathSciNetzbMATHGoogle Scholar
  3. Caporossi G, Hansen P (2000) Variable neighborhood search for extremal graphs: 1 The AutoGraphiX system. Discrete Math 212(1–2):29–44MathSciNetCrossRefzbMATHGoogle Scholar
  4. Dobrynin AA, Entringer RC, Gutman I (2001) Wiener index of trees: theory and Applications. Acta Appl Math 6(6):211–249MathSciNetCrossRefzbMATHGoogle Scholar
  5. Eliasi M (2009) Harary index of aigzag polyhex nanotorus. Dig J Nanomater Biostruct 4(4):757–762Google Scholar
  6. Fu HY, Xie DZ (2010) Equitable \(L(2,1)\)-labelings of Sierpiński graphs. Australas J Comb 46:147–156zbMATHGoogle Scholar
  7. Gravier S, Klavžar S, Mollard M (2005) Codes and \(L(2,1)\)-labelings in Sierpiński-like graphs. Taiwan J Math 9:671–681CrossRefzbMATHGoogle Scholar
  8. Gutman I, Vidović D, Popović L (1998) Graph representation of organic molecules Cayley’s plerograms vs. his kenograms. J Chem Soc Faraday Trans 94(7):875-860CrossRefGoogle Scholar
  9. Gutman I (1997) A property of the Wiener number and its modifications. Indian J Chem 36A:128–132Google Scholar
  10. Gravier S, Klavžar S, Mollard M (2005) Codes and \(L(2,1)\)-labelings in Sierpiński-like graphs. Taiwan J Math 9:671–681CrossRefzbMATHGoogle Scholar
  11. Hinz AM (1992) Pascal’s triangle and the Tower of Hanoi. Am Math Mon 99:538–544MathSciNetCrossRefzbMATHGoogle Scholar
  12. Hinz AM, Klavžar S, Milutinović U, Parisse D, Petr C (2005) Metric properties of the Tower of Hanoi graphs and Stern’s diatomic sequence. Eur J Comb 26:693–708MathSciNetCrossRefzbMATHGoogle Scholar
  13. Hinz AM, Parisse D (2012) Coloring Hanoi and Sierpiński graphs. Discrete Math 312:1521–1535MathSciNetCrossRefzbMATHGoogle Scholar
  14. Hinz AM, Parisse D (2012) The average eccentricity of Sierpiński graphs. Graphs Comb 28(5):1–16CrossRefzbMATHGoogle Scholar
  15. Jakavac M, Klavžar S (2009) Vertex-, edge-, and total-coloring of Sierpiński-like graphs. Discrete Math 309:1548–1556MathSciNetCrossRefzbMATHGoogle Scholar
  16. Jakovac M (2014) A 2-parametric generalization of Sierpiński gasket graphs. Ars Comb 116(3):395–405MathSciNetzbMATHGoogle Scholar
  17. Klavžar S, Milutinović U (1997) Graphs \(S(n, k)\) and a variant of the Tower of Hanoi problem. J Czechoslov Math 47(122):95–104MathSciNetCrossRefzbMATHGoogle Scholar
  18. Klavžar S, Mohar B (2005) Crossing numbers of Sierpiński-like graphs. J Graph Theory 50:186–198MathSciNetCrossRefzbMATHGoogle Scholar
  19. Klavžar S, Milutinović U, Petr C (2002) 1-Perfect codes in Sierpiński-like graphs. Bull Aust Math Soc 66:369–384CrossRefzbMATHGoogle Scholar
  20. Klavžar S, Zemljic SS (2013) On distances in Sierpinski graphs: almost- extreme vertices and metric dimension. Appl Anal Discrete Math 7(1):72–82MathSciNetCrossRefzbMATHGoogle Scholar
  21. Klein DJ, Lukovits I, Gutman I (1995) On the definition of the hyper-Wiener index for cycle-containing structures. J Chem Inf Comput Sci 35:50–52CrossRefGoogle Scholar
  22. Lin CH, Liu JJ, Wang YL, Yen WC (2011) The hub number of Sierpiński-like graphs. Theory Comput Syst 49:588–600MathSciNetCrossRefzbMATHGoogle Scholar
  23. Lipscomb SL, Perry JC (1992) Lipscomb’s \(L(A)\) space fractalized in Hilbert’s \(l^2(A)\) space. Proc Am Math Soc 115:1157–1165MathSciNetzbMATHGoogle Scholar
  24. Luo C, Zuo L (2017) The metric properties of Sierpiński-like graphs. Appl Math Comput 296:124–136MathSciNetGoogle Scholar
  25. Milutinović U (1992) Completeness of the Lipscomb space. Glas Ser III 27(47):343–364MathSciNetzbMATHGoogle Scholar
  26. Parisse D (2009) On some metric properties of the Sierpiński-like graphs \(S(n, k)\). Ars comb 90:145–160MathSciNetzbMATHGoogle Scholar
  27. Pisanski T, Tucker TW (2001) Growth in repeated truncations of maps. Atti Sem Mat Fis Univ Modena 49:167–176MathSciNetzbMATHGoogle Scholar
  28. Romik D (2006) Shortest paths in the Tower of Hanoi graph and finite automata. SIAM J Discrete Math 20:610–622MathSciNetCrossRefzbMATHGoogle Scholar
  29. Trinajstić N, Li X, Gutman I (2006) Mathematical aspects of Randi\(\acute{c}\)-type molecular structure desriptors. Croat Chem Acta 79(3):A31–A32Google Scholar
  30. Xue B, Zuo L, Li GJ (2012) The hamiltonicity and path \(t\)-coloring of Sierpiński-like graphs. Discrete Appl Math 160:1822–1836MathSciNetCrossRefzbMATHGoogle Scholar
  31. Xue B, Zuo L, Wang G, Li GJ (2015) The linear \(t\)-colorings of Sierpiński-like graphs. Graphs Comb 31:1795–1805CrossRefzbMATHGoogle Scholar
  32. Zhou B, Trinajstić N (2009) On a novel connectivity index. J Math Chem 46(4):1252–1270MathSciNetCrossRefzbMATHGoogle Scholar
  33. Zhou B, Trinajstić N (2010) On general sum-connectivity index. J Math Chem 47(1):210–218MathSciNetCrossRefzbMATHGoogle Scholar
  34. Zuo L (2012) About a conjecture on the Randi\(\acute{c}\) Index of graphs. Bull Malays Math Soc 2(2):411–424MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.College of Mathematical ScienceTianjin Normal UniversityTianjinChina

Personalised recommendations