Graphs and Combinatorics

, Volume 28, Issue 5, pp 671–686 | Cite as

The Average Eccentricity of Sierpiński Graphs

  • Andreas M. HinzEmail author
  • Daniele Parisse
Original Paper


We determine the eccentricity of an arbitrary vertex, the average eccentricity and its standard deviation for all Sierpiński graphs \({S_p^n}\). Special cases are the graphs \({S_2^{n}}\), which are isomorphic to the state graphs of the Chinese Rings puzzle with n rings and the graphs \({S_3^{n}}\) isomorphic to the Hanoi graphs \({H_3^{n}}\) representing the Tower of Hanoi puzzle with 3 pegs and n discs.


Sierpiński graphs Eccentricity Average eccentricity Tower of Hanoi Hanoi graphs Integer sequences 

Mathematics Subject Classification (2010)

05C12 05A10 11B83 11B73 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Afriat S.N.: The Ring of Linked Rings. Duckworth, London (1982)Google Scholar
  2. 2.
    Buckley F., Harary F.: Distance in Graphs. Addison-Wesley, Redwood City (1990)zbMATHGoogle Scholar
  3. 3.
    Dankelmann P., Goddard W., Swart C.S.: The average eccentricity of a graph and its subgraphs. Util. Math. 65, 41–51 (2004)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Graham R.L., Knuth D.E., Patashnik O.: Concrete Mathematics, 2nd edn. Addison-Wesley, Reading (1994)zbMATHGoogle Scholar
  5. 5.
    Hinz A.M.: The Tower of Hanoi. Enseign. Math. (2) 35, 289–321 (1989)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Hinz A.M.: Pascal’s triangle and the Tower of Hanoi. Am. Math. Monthly 99, 538–544 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Hinz, A.M.: The Tower of Hanoi. In: Shum, K.P., Taft, E.J., Wan, Z.X. (eds.) Algebras and Combinatorics. An International Congress, ICAC ’97, Hong Kong, pp. 277–289. Springer, Singapore (1999)Google Scholar
  8. 8.
    Hinz, A.M.: Graph theory of tower tasks. Behavioural Neurology (to appear, 2011)Google Scholar
  9. 9.
    Hinz A. M., Klavžar S., Milutinović U., Parisse D., Petr C.: Metric properties of the Tower of Hanoi graphs and Stern’s diatomic sequence. Eur. J. Combin. 26, 693–708 (2005)CrossRefGoogle Scholar
  10. 10.
    Hinz A.M., Kostov A., Kneißl F., Sürer F., Danek A.: A mathematical model and a computer tool for the Tower of Hanoi and Tower of London puzzles. Inform. Sci. 179, 2934–2947 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Hinz, A.M., Parisse, D.: Coloring Hanoi and Sierpiński graphs. Discrete Math. (to appear, 2011)Google Scholar
  12. 12.
    Hinz A.M., Schief A.: The average distance on the Sierpiński gasket. Probab. Theory Relat. Fields 87, 129–138 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Jakovac M., Klavžar S.: Vertex-, edge-, and total colorings of Sierpiński-like graphs. Discrete Math. 309, 1548–1556 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Klavžar S.: Coloring Sierpiński graphs and Sierpiński gasket graphs. Taiwan J. Math. 12, 513–522 (2008)zbMATHGoogle Scholar
  15. 15.
    Klavžar S., Milutinović U.: Graphs S(n, k) and a variant of the Tower of Hanoi problem. Czechoslov. Math. J. 47(122), 95–104 (1997)zbMATHCrossRefGoogle Scholar
  16. 16.
    Klavžar S., Milutinović U., Petr C.: 1-perfect codes in Sierpiński graphs. Bull. Austral. Math. Soc. 66, 369–384 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Klavžar S., Mohar B.: Crossing numbers of Sierpiński-like graphs. J. Graph Theory 50, 186–198 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Korf R. Best-first frontier search with delayed duplicate detection. In: McGuinness, D.L., Ferguson, G. (eds) Proceedings of the Nineteenth National Conference on Artificial Intelligence (AAAI-2004), pp. 650–657. AAAI Press/The MIT Press, Menlo Park/Cambridge (2004)Google Scholar
  19. 19.
    Parisse D.: The Tower of Hanoi and the Stern-Brocot Array. Thesis, München (1997)Google Scholar
  20. 20.
    Parisse D.: On some metric properties of the Sierpiński graphs S(n, k). Ars Comb. 90, 145–160 (2009)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Romik D.: Shortest paths in the Tower of Hanoi graph and finite automata. SIAM J. Discrete Math. 20, 610–622 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Sloane’s Online Encyclopedia of Integer Sequences. (2011–07–26)
  23. 23.
    Wiesenberger, H.L.: Stochastische Eigenschaften von Hanoi- und Sierpiński-Graphen. Thesis of Diploma, München (2010)Google Scholar

Copyright information

© Springer 2011

Authors and Affiliations

  1. 1.FernUniversität in HagenHagenGermany
  2. 2.EADS Deutschland GmbHMunichGermany

Personalised recommendations