Abstract
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 edgea∈A/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.
Similar content being viewed by others
References
N. Deo,Graph Theory with Applications to Engineering and Computer Science, Printice-Hall, New Jersey, 1974.
L. R. Ford andD. R. Fulkerson,Flows in Networks, Princeton University Press, New Jersey, 1962.
S. L. Hakimi andW. Maeda, On coefficients of polynomials in network function,IRE Trans. Circuit Theory, CT-7 (1960), 40–44.
P. Hall, On representatives of subsets,J. London Math. Soc.,10 (1935), 26–30.
Y. Kajitani, Graph theoretical properties of the node determinant of an LCR network,IEEE Trans. Circuit Theory, CT-18 (1971), 343–350.
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.
J. B. Kruskal, On the shortest spanning subtree of a graph and the traveling salesman problem,Proc. Amer. Math. Soc.,7 (1956), 48–50.
E. Okamoto andY. Kajitani, A class of trees that determines a weighted graph (in Japanese),Trans. IECE of Japan,61-A (1978), 604–610.
R. C. Prim, Shortest connection networks and some generalizations,Bell System Tech. J.,36 (1957), 1389–1401.
Author information
Authors and Affiliations
Rights 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). https://doi.org/10.1007/BF02579450
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02579450