Skip to main content
Springer Nature Link
Log in

Spanning tree formulas and chebyshev polynomials

  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

The Kirchhoff Matrix Tree Theorem provides an efficient algorithm for determiningt(G), the number of spanning trees of any graphG, in terms of a determinant. However for many special classes of graphs, one can avoid the evaluation of a determinant, as there are simple, explicit formulas that give the value oft(G). In this work we show that many of these formulas can be simply derived from known properties of Chebyshev polynomials. This is demonstrated for wheels, fans, ladders, Moebius ladders, and squares of cycles. The method is then used to derive a new spanning tree formula for the complete prismR n (m) =K m ×C n . It is shown that

$$2^{\left( {\begin{array}{*{20}c} n \\ 2 \\ \end{array} } \right)\left( {1 - \frac{1}{{r - 1}} + o\left( 1 \right)} \right)} $$

whereT n (x) is then th order Chebyshev polynomial of the first kind.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Baron, G., Boesch, F., Prodinger, H., Tichy, R., Wang, J.: The number of spanning trees in the square of a cycle. Fibonacci Q. (to appear)

  2. Bellman, R.: Introduction to Matrix Analysis. New York: McGraw Hill 1970

    Google Scholar 

  3. Biggs, N.: Algebraic Graph Theory. London: Cambridge University Press 1974

    Google Scholar 

  4. Boesch, F., Bogdanowicz, Z.: The number of spanning trees in a prism. Stevens Institute Computer Science Report 8405 (1984)

  5. Boesch, F.T., Wang, J.F.: A conjecture on the number of spanning trees in the square of a cycle. In: Notes from New York Graph Theory Day V, p. 16. New York: New York Academy Sciences 1982

    Google Scholar 

  6. Feussner, W.: Zur Berechnung der Stromsträrke in netzförmigen Leitern. Ann. Phys.15, 385–394 (1904)

    Google Scholar 

  7. Harry, F.: Graph Theory. Reading: Addison-Wesley 1969

    Google Scholar 

  8. Hilton, A.J.W.: Spanning trees and Fibonacci and Lucas numbers. Fibonacci Q.12, 259–262 (1974)

    Google Scholar 

  9. Kel'mans, A.K., Chelnokov, V.M.: A certain polynomial of a graph and graphs with an extremal number of trees. J. Comb. Theory (B)16, 197–214 (1974)

    Google Scholar 

  10. Kirchhoff, G.: 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)

    Google Scholar 

  11. Kleitman, D.J., Golden, B.: Counting trees in a certain class of graphs. Amer. Math. Mon.82, 40–44 (1975)

    Google Scholar 

  12. Marcus, M., Minc, H.: A Survey of Matrix Theory and Matrix Inequalities. Boston: Allyn and Bacon 1964

    Google Scholar 

  13. Moon, J.W. Counting Labelled Trees. Canadian Mathematical Congress Monograph 1. London: William Clowes & Sons 1970

    Google Scholar 

  14. Sachs, H.: Über selbstkomplementäre Graphen. Publ. Math. Debrecen9, 270–288 (1962)

    Google Scholar 

  15. Schwenk, A., Wilson, R.: On the eigenvalues of a graph. In: Selected Topics in Graph Theory, edided by Beineke, Wilson, pp. 307–336. New York: Academic Press 1978

    Google Scholar 

  16. Schwenk, A.: Computing the characteristic polynomial of a graph. In: Graphs and Combinatorics, Lecture Notes in Mathematics406, pp. 153–172. Berlin-Heidelberg-New York: Springer-Verlag 1974

    Google Scholar 

  17. Sedlácěk, J.: Lucas numbers in graph theory. In: Mathematics-Geometry and Graph Theory (in Czechoslovak), pp. 111–115. Prague: Univ. Karlova 1970. See also Sedlácěk, J.: Ungerichtete Graphen und ihre Gerüste. In: Beiträge zur Graphen Theorie, edited by (H. Sachs, H.-J. Vosz, H. Walther, pp. 143–146. Leipzig: Teubner 1968

    Google Scholar 

  18. Sedlácěk, J.: On the spanning trees of finite graphs. Cas. Pestovani Mat.94, 217–221 (1969)

    Google Scholar 

  19. Sedlácěk, J.: On the skeletons of a graph or digraph. In: Combinatorial Structures and their Applications, editded by R. Guy, M. Hanani, N. Saver, J. Schonheim, pp. 387–391. New York: Gordon and Breach 1970

    Google Scholar 

  20. Snyder, M.: Chebyshev Methods in Numerical Analysis. Englewood Cliffs: Prentice-Hall 1966

    Google Scholar 

  21. Temperley, H.: On the mutual cancellation of cluster integrals in Mayer's fugacity series. Proc. Phys. Soc. Lond. Ser. A83, 3–16 (1964)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Computer Science Department, Stevens Institute of Technology, Hoboken, NJ, USA

    F. T. Boesch

  2. Institut für Algebra und Diskrete Mathematik, Technische Universität, Wien, Austria

    H. Prodinger

Authors
  1. F. T. Boesch
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. H. Prodinger
    View author publications

    You can also search for this author inPubMed Google Scholar

Additional information

The work of this author was supported under NSF Grant ECS-8100652.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Boesch, F.T., Prodinger, H. Spanning tree formulas and chebyshev polynomials. Graphs and Combinatorics 2, 191–200 (1986). https://doi.org/10.1007/BF01788093

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01788093

Keywords