A182258 Least number k such that there exists a simple graph on k vertices having precisely n spanning trees.

3, 4, 5, 6, 7, 4, 5, 10, 5, 5, 13, 6, 6, 4, 7, 8, 7, 5, 5, 22, 8, 5, 9, 8, 7, 6, 6, 6, 9, 6, 7, 10, 6, 6, 7, 10, 7, 5, 7, 8, 7, 7, 5, 7, 11, 6, 7, 7, 7, 6, 8, 6, 6, 6, 8, 8, 8, 6, 6, 9, 7, 6, 8, 6, 8, 7, 6, 8, 9, 7, 7, 9, 5, 7, 9, 9, 7, 7, 6

Offset

3,1

Comments

The only fixed points for a(n) are 3, 4, 5, 6, 7, 10, 13, 22.

If n > 25 and n != 2 (mod 3) then a(n) <= (n+9)/4.

If n > 5 and n = 2 (mod 3) then a(n) <= (n+4)/3. [corrected by Jukka Kohonen, Feb 16 2022]

It is conjectured that a(n) = o(log(n)).

a(mn) <= a(m)+a(n)-1, by joining two graphs with m and n spanning trees at a single common vertex. - Jukka Kohonen, Feb 17 2022

Links

Jukka Kohonen, Table of n, a(n) for n = 3..10000

Jernej Azarija, MathOverflow: Minimal graphs with a prescribed number of spanning trees

Jernej Azarija and Riste Škrekovski, Euler's idoneal numbers and an inequality concerning minimal graphs with a prescribed number of spanning trees, Mathematica Bohemica, Vol. 138 (2013), No. 2, 121--131.

Dick Lipton, The Inverse Spanning Tree Problem

Ladislav Nebeský, On the minimum number of vertices and edges in a graph with a given number of spanning trees, Časopis pro pěstování matematiky 98 (1973), 95-97.

J. Sedláček, On the minimal graph with a given number of spanning trees, Canad. Math. Bull. 13 (1970) 515-517.

Example

a(100000000) = 10 since K_10 has 100000000 spanning trees.
From Jukka Kohonen, Feb 17 2022: (Start)
a(47) = 11 since the following graph has 47 spanning trees:
    o-o-o-o
   /       \
  o--o---o--o
   \       /
    o--o--o
(End)

Crossrefs

Sequence in context: A111608 A126800 A245689 * A067628 A168093 A095254

Adjacent sequences: A182255 A182256 A182257 * A182259 A182260 A182261

Keyword

nonn

Author

Jernej Azarija, Apr 21 2012

Extensions

a(14)-a(46) from Jukka Kohonen, Feb 16 2022

a(47)-a(81) from Jukka Kohonen, Feb 17 2022

Status

approved