Spanning trees of extended graphs

Abstract

A generalization of the Prüfer coding of trees is given providing a natural correspondence between the set of codes of spanning trees of a graph and the set of codes of spanning trees of theextension of the graph. This correspondence prompts us to introduce and to investigate a notion ofthe spanning tree volume of a graph and provides a simple relation between the volumes of a graph and its extension (and in particular a simple relation between the spanning tree numbers of a graph and its uniform extension). These results can be used to obtain simple purely combinatorial proofs of many previous results obtained by the Matrix-tree theorem on the number of spanning trees of a graph. The results also make it possible to construct graphs with the maximal number of spanning trees in some classes of graphs.

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References

  1. [1]

    J. A. Bondy, andU. S. R. Murty:Graph Theory with Applications, New York: Elsevier, London: MacMillan, 1976.

    Google Scholar 

  2. [2]

    A. K. Kelmans: The number of spanning trees of a graph containing a given forestActa Math. Acad. Sci. Hungar.,27 (1976), 89–95.

    Google Scholar 

  3. [3]

    A. K. Kelmans: The number of trees of a graph I, IIAvtomat. i Telemekh,3, (1965), 2194–2204;2, (1966), 56–65. (English transl. in Automat. Remote Control26 (1965);27 (1966)).

    Google Scholar 

  4. [4]

    A. K. Kelmans: Operations on graphs that increase the number of their spanning trees, in: Issledovaniya po Diskretnoy optimizacii, Nauka, Moskva 1976, 406–424. (in Russian, MR 56 #3342).

    Google Scholar 

  5. [5]

    A. K. Kelmans: Comparison of graphs by their number of spanning trees,Discrete Mathematics 16 (1976), 241–261.

    Google Scholar 

  6. [6]

    A. K. Kelmans, andV. N. Chelnokov: A certain polynomial of a graph and graphs with extremal number of trees,J. Combinatorial Theory, (B)16 (1974), 197–214.

    Google Scholar 

  7. [7]

    Gy. Oláh: A problem on the number of trees,Studia Sci. Math. Hungar. 3 (1968), 71–80.

    Google Scholar 

  8. [8]

    A. Prüfer: Neuer Beweis eines Satzes über Permutationen,Archiv für Math. u. Phys. 27 (1918), 142–144.

    Google Scholar 

  9. [9]

    A. Rényi: Új módszerek és eredmények a kombinatorikus analízisben I,Magyar Tud. Akad. Mat. Fiz. Oszt. Közl. 16 (1966), 77–105.

    Google Scholar 

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Kelmans, A.K. Spanning trees of extended graphs. Combinatorica 12, 45–51 (1992). https://doi.org/10.1007/BF01191204

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AMS subject classification code (1991)

  • 05 C 05
  • 05 C 30