Spanning trees of extended graphs


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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Kelmans, A.K. Spanning trees of extended graphs. Combinatorica 12, 45–51 (1992).

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AMS subject classification code (1991)

  • 05 C 05
  • 05 C 30