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

TL;DR

This paper investigates bounds on the minimal number of vertices and edges in graphs with a given number of spanning trees, establishing new inequalities related to Euler's idoneal numbers.

Contribution

It introduces improved bounds on minimal graph sizes for prescribed spanning trees, linking these bounds to Euler's idoneal numbers and providing necessary and sufficient conditions.

Findings

Established that (n)    (n+4)/3 for certain n

Established that (n)    (n+7)/3 for certain n

Provided refined bounds for (n) and (n) under specific modular conditions

Abstract

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.