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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. 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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. 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