Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

27
The categories of this paper are: comp-gas
5
The categories of this paper are: gr-qc
0
The categories of this paper are: hep-ph
25
The categories of this paper are: patt-sol
23
The categories of this paper are: solv-int
37
The categories of this paper are: supr-con
13
The categories of this paper are: quant-ph
7
The categories of this paper are: hep-th
22
The categories of this paper are: chao-dyn
35
The categories of this paper are: mtrl-th
3
The categories of this paper are: physics
33
The categories of this paper are: plasm-ph
34
The categories of this paper are: chem-ph
21
The categories of this paper are: adap-org
9
The categories of this paper are: nlin
24
The categories of this paper are: dg-ga