![](/files/happy5.png)
![$ n \geq 3 $](/files/tex/03699c2971c6eb403dcc4c26a1c9818ad7b45da8.png)
![$ \alpha(n) $](/files/tex/7cf5e54f55a113042ef4e18f16bdf6601a27462e.png)
![$ k $](/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png)
![$ k $](/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png)
![$ n $](/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png)
![$ \alpha(n) = o(\log{n}). $](/files/tex/196d4bae3ba249904f8319dc18b830afd9cc7fe8.png)
Observe that is well defined for
since
has
spanning trees.
The function was introduced by Sedlacek [S] who has shown that for large enough
and
Using the fact that almost all positive integers are expressible as
for integers
it can be shown [A] that for large enough
and
otherwise.
Moreover, the only fixed points of are 3, 4, 5, 6, 7, 10, 13 and 22.
The conjecture is motivated by the following graph (ploted for a very small sample of vertices)
The conjecture [C] is justifiable for highly composite numbers since in this case one can construct the graph obtained after taking cycles
for every odd prime factor
of
.
Bibliography
[S] J. Sedlacek, On the minimal graph with a given number of spanning trees, Canad. Math. Bull. 13 (1970) 515-517.
[A] J. Azarija, R. Skrekovski, Euler's idoneal numbers and an inequality concerning minimal graphs with a prescribed number of spanning trees, IMFM preprints 49 (2011) Link to paper
* [C] Minimal graphs with a prescribed number of spanning trees
* indicates original appearance(s) of problem.