Annals of Combinatorics

, 15:355 | Cite as

The Number of Spanning Trees in Self-Similar Graphs

  • Elmar TeuflEmail author
  • Stephan Wagner


The number of spanning trees of a graph, also known as the complexity, is computed for graphs constructed by a replacement procedure yielding a self-similar structure. It is shown that under certain symmetry conditions exact formulas for the complexity can be given. These formulas indicate interesting connections to the theory of electrical networks. Examples include the well-known Sierpiński graphs and their higher-dimensional analogues. Several auxiliary results are provided on the way—for instance, a property of the number of rooted spanning forests is proven for graphs with a high degree of symmetry.

Mathematics Subject Classification

05C30 05C05 34B45 


spanning trees self-similar graphs 


  1. 1.
    Barlow M.T.: Diffusions on fractals. In: Bernard, P. (ed.) Lectures on Probability Theory and Statistics, pp. 1–121. Springer, Berlin (1998)CrossRefGoogle Scholar
  2. 2.
    Berge C.: Graphs and Hypergraphs. North-Holland Publishing Co., Amsterdam (1976)zbMATHGoogle Scholar
  3. 3.
    Bollobás B.: Modern Graph Theory. Graduate Texts in Mathematics, Vol. 184. Springer-Verlag, New York (1998)Google Scholar
  4. 4.
    Brown T.J.N., Mallion R.B., Pollak P., Roth A.: Some methods for counting the spanning trees in labelled molecular graphs, examined in relation to certain fullerenes. Discrete Appl. Math. 67(1-3), 51–66 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Cayley A.: A theorem on trees. Quart. J. Math. 23, 376–378 (1889)Google Scholar
  6. 6.
    Chaiken S.: A combinatorial proof of the all minors matrix tree theorem. SIAM J. Algebraic Discrete Methods 3(3), 319–329 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Chang S.-C., Chen L.-C., Yang W.-S.: Spanning trees on the Sierpinski gasket. J. Stat. Phys. 126(3), 649–667 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Colbourn C.J.: The Combinatorics of Network Reliability. Oxford University Press, New York (1987)Google Scholar
  9. 9.
    Guido D., Isola T., Lapidus M.L.: A trace on fractal graphs and the Ihara zeta function. Trans. Amer. Math. Soc. 361(6), 3041–3070 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Harary F., Palmer E.M.: Graphical Enumeration. Academic Press, New York (1973)zbMATHGoogle Scholar
  11. 11.
    Kigami J.: Analysis on Fractals. Cambridge Tracts in Mathematics, Vol. 143. Cambridge University Press, Cambridge (2001)Google Scholar
  12. 12.
    Kirchhoff G.R.: Über die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Ströme geführt wird. Ann. Phys. Chem. 72, 497–508 (1847)CrossRefGoogle Scholar
  13. 13.
    Krön B.: Growth of self-similar graphs. J. Graph Theory 45(3), 224–239 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Lyons R.: Asymptotic enumeration of spanning trees. Combin. Probab. Comput. 14(4), 491–522 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Metz V.: The short-cut test. J. Funct. Anal. 220(1), 118–156 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Moon J.W.: Some determinant expansions and the matrix-tree theorem. Discrete Math. 124(1-3), 163–171 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Neunhäuserer J.: Random walks on infinite self-similar graphs. Electron. J. Probab. 12(46), 1258–1275 (2007)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Sabot, C.: Spectral properties of self-similar lattices and iteration of rational maps. Mém. Soc. Math. Fr. (N.S.) 92, (2003)Google Scholar
  19. 19.
    Shima T.: On eigenvalue problems for Laplacians on p.c.f. self-similar sets. Japan J. Indust. Appl. Math. 13(1), 1–23 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Shrock R., Wu F.Y.: Spanning trees on graphs and lattices in d dimensions. J. Phys. A 33(21), 3881–3902 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Teufl, E.,Wagner, S.: The number of spanning trees of finite Sierpiński graphs. In: Fourth Colloquium on Mathematics and Computer Science, pp. 411–414. Nancy (2006)Google Scholar
  22. 22.
    Teufl E., Wagner S.: Enumeration problems for classes of self-similar graphs. J. Combin. Theory Ser. A 114(7), 1254–1277 (2007)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Fachbereich MathematikEberhard Karls Universität TübingenTübingenGermany
  2. 2.Department of Mathematical SciencesStellenbosch UniversityMatielandSouth Africa

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