Euler's idoneal numbers and an inequality concerning minimal graphs with a prescribed number of spanning trees

Let $\alpha(n)$ be the least number $k$ for which there exists a simple graph with $k$ vertices having precisely $n \geq 3$ spanning trees. Similarly, define $\beta(n)$ as the least number $k$ for which there exists a simple graph with $k$ edges having precisely $n \geq 3$ spanning trees. As an $n$-cycle has exactly $n$ spanning trees, it follows that $\alpha(n),\beta(n) \leq n$. In this paper, we show that $\alpha(n) \leq \frac{n+4}{3}$ and $\beta(n) \leq \frac{n+7}{3} $ if and only if $n \notin {3,4,5,6,7,9,10,13,18,22}$, which is a subset of Euler's idoneal numbers. Moreover, if $n \not \equiv 2 \pmod{3}$ and $n \not = 25$ we show that $\alpha(n) \leq \frac{n+9}{4}$ and $\beta(n) \leq \frac{n+13}{4}.$ This improves some previously known bounds.