Minimal graphs with a prescribed number of spanning trees. (English) Zbl 1490.05127
Summary: For \(n\geq 3\), let \(\alpha(n)\) denote the minimum number of vertices in a graph with exactly \(n\) spanning subtrees. This notion was introduced by J. Sedláček [Can. Math. Bull. 13, 515–517 (1970; Zbl 0202.23501)] and has been studied by J. Azarija and R. Škrekovski [Math. Bohem. 138, No. 2, 121–131 (2013; Zbl 1289.05043)] In particular, they have conjectured that \(\alpha(n)=o(\log n)\). This paper will prove a weak version of this conjecture: specifically we will show that \(\alpha(n)=O((\log n)^{3/2}/(\log\log n))\). This bound is substantially larger than the conjectured upper bound; it is at least in the same ballpark.
MathOverflow Questions:
Does the quadratic form \(x^2 - 7y^2\) represent infinitely many primes, with the restriction that \(0 < y < x/10\)?Hecke equidistribution
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MathOverflowReferences:
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| [4] | Open Problem Garden, Minimal graphs with a prescribed number of spanning trees,http://garden.irmacs.sfu.ca/category/graph_theory, Accessed: 2022-01-04. · Zbl 0781.52005 |
| [5] | J. Sedl´a˘cek, On the minimal graph with a given number of spanning trees,Canad. Math. Bull.13 (1970), 515-517. · Zbl 0202.23501 |
| [6] | https://mathoverflow.net/questions/133410/hecke-equidistribution; Accessed: 2022-01-04. |
| [7] | https://mathoverflow.net/questions/65059/ does-the-quadratic-form-x2-7y2-represent-infinitely-many-primes-with-; Accessed: 2022-01-04 |
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