Maximum andk-th maximal spanning trees of a weighted graph


LetA be a maximum spanning tree andP be an arbitrary spanning tree of a connected weighted graphG. Then we prove that there exists a bijectionψ fromA/P intoP/A such that for any edgeaA/P, (P/ψ(a)) ∪a is a spanning tree ofG whose weight is greater than or equal to that ofP. We apply this theorem to some problems concerning spanning trees of a weighted graph.

This is a preview of subscription content, access via your institution.


  1. [1]

    N. Deo,Graph Theory with Applications to Engineering and Computer Science, Printice-Hall, New Jersey, 1974.

    Google Scholar 

  2. [2]

    L. R. Ford andD. R. Fulkerson,Flows in Networks, Princeton University Press, New Jersey, 1962.

    Google Scholar 

  3. [3]

    S. L. Hakimi andW. Maeda, On coefficients of polynomials in network function,IRE Trans. Circuit Theory, CT-7 (1960), 40–44.

    Google Scholar 

  4. [4]

    P. Hall, On representatives of subsets,J. London Math. Soc.,10 (1935), 26–30.

    MATH  Article  Google Scholar 

  5. [5]

    Y. Kajitani, Graph theoretical properties of the node determinant of an LCR network,IEEE Trans. Circuit Theory, CT-18 (1971), 343–350.

    MathSciNet  Article  Google Scholar 

  6. [6]

    T. Kawamoto, Y. Kajitani andS. Shinoda, On the second maximal trees of a weighted graph (in Japanese).Trans. IECE of Japan,61-A (1978), 988–995.

    MathSciNet  Google Scholar 

  7. [7]

    J. B. Kruskal, On the shortest spanning subtree of a graph and the traveling salesman problem,Proc. Amer. Math. Soc.,7 (1956), 48–50.

    Article  MathSciNet  Google Scholar 

  8. [8]

    E. Okamoto andY. Kajitani, A class of trees that determines a weighted graph (in Japanese),Trans. IECE of Japan,61-A (1978), 604–610.

    Google Scholar 

  9. [9]

    R. C. Prim, Shortest connection networks and some generalizations,Bell System Tech. J.,36 (1957), 1389–1401.

    Google Scholar 

Download references

Author information



Rights and permissions

Reprints and Permissions

About this article

Cite this article

Kano, M. Maximum andk-th maximal spanning trees of a weighted graph. Combinatorica 7, 205–214 (1987).

Download citation

AMS subject classification (1980)

  • 05 C 05