Sharp estimates for spanning trees

Steven Klee; Bhargav Narayanan; Lisa Sauermann

Sharp estimates for spanning trees

Combinatorics 2022-04-14 v4

Authors: Steven Klee , Bhargav Narayanan , Lisa Sauermann

Abstract

We prove the following sharp estimate for the number of spanning trees of a graph in terms of its vertex-degrees: a simple graph $G$ on $n$ vertices has at most $(1/n^{2}) \prod_{v \in V(G)} (d(v)+1)$ spanning trees. This result is tight (for complete graphs), and improves earlier estimates of Alon from 1990 and Kostochka from 1995 by a factor of about $1/n$ (for dense graphs). We additionally show that an analogous bound holds for the weighted spanning tree enumerator of a (nonnegatively) weighted graph as well.

Keywords

spanning trees trees graph theory

Cite

@article{arxiv.2102.01669,
  title  = {Sharp estimates for spanning trees},
  author = {Steven Klee and Bhargav Narayanan and Lisa Sauermann},
  journal= {arXiv preprint arXiv:2102.01669},
  year   = {2022}
}

Comments

It has been pointed out to us when this paper was under review that the results are not new; in fact a stronger bound was proved by Grone and Merris [Discrete Math. 69 (1988), 97-99]

Sharp estimates for spanning trees

Abstract

Keywords

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