Geodesic
Counting graphs with different numbers of spanning trees through the counting of prime partitions
Czechoslovak Mathematical Journal, Tome 64 (2014) no. 1, pp. 31-35 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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Let $A_n$ $(n \geq 1)$ be the set of all integers $x$ such that there exists a connected graph on $n$ vertices with precisely $x$ spanning trees. It was shown by Sedláček that $|A_n|$ grows faster than the linear function. In this paper, we show that $|A_{n}|$ grows faster than $\sqrt {n} {\rm e}^{({2\pi }/{\sqrt 3})\sqrt {n/\log {n}}}$ by making use of some asymptotic results for prime partitions. The result settles a question posed in J. Sedláček, On the number of spanning trees of finite graphs, Čas. Pěst. Mat., 94 (1969), 217–221.
Let $A_n$ $(n \geq 1)$ be the set of all integers $x$ such that there exists a connected graph on $n$ vertices with precisely $x$ spanning trees. It was shown by Sedláček that $|A_n|$ grows faster than the linear function. In this paper, we show that $|A_{n}|$ grows faster than $\sqrt {n} {\rm e}^{({2\pi }/{\sqrt 3})\sqrt {n/\log {n}}}$ by making use of some asymptotic results for prime partitions. The result settles a question posed in J. Sedláček, On the number of spanning trees of finite graphs, Čas. Pěst. Mat., 94 (1969), 217–221.
DOI : 10.1007/s10587-014-0079-8
Classification : 05A16, 05C35
Keywords: number of spanning trees; asymptotic; prime partition 
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Azarija, Jernej. Counting graphs with different numbers of spanning trees through the counting of prime partitions. Czechoslovak Mathematical Journal, Tome 64 (2014) no. 1, pp. 31-35. doi: 10.1007/s10587-014-0079-8

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