Abstract
Let A n (n ⩽ 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 \(\sqrt n e^{(2\pi /\sqrt 3 )\sqrt {n/\log n} } \) 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.
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Azarija, J. Counting graphs with different numbers of spanning trees through the counting of prime partitions. Czech Math J 64, 31–35 (2014). https://doi.org/10.1007/s10587-014-0079-8
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DOI: https://doi.org/10.1007/s10587-014-0079-8