A182290 Maximal number of connected graphs of order n having distinct numbers of spanning trees.

1, 1, 2, 5, 16, 65, 386, 3700, 55784, 1134526, 27053464

Offset: 1

Author

Jernej Azarija, Jun 27 2012

Keywords

• nonn
• hard
• more

Comments

a(n) grows asymptotically faster than sqrt(n)*exp(2*Pi*sqrt(n/log(n))/sqrt(3)).

Examples

			a(3) = 2 since any connected graph on 3 vertices can either have 1 spanning tree (any tree) or 3 (triangle).
		

Links

• Jernej Azarija, Counting graphs with different numbers of spanning trees through the counting of prime partitions (preprint, 2012).
• Jernej Azarija, Maximal class of simple graphs of order n with mutually distinct number of spanning trees, (Mathoverflow).
• J. Sedlacek, On the number of spanning trees of finite graphs, Cas. Pro. Pest Mat., Vol. 94 (1969) 217-221.

Programs

Sage

# needs the package nauty:
len( set([g.spanning_trees_count() for g in graphs.nauty_geng('-c ' + str(n)) ]))

Extensions

a(11) from Jernej Azarija, Sep 07 2012