Counting graphs with different numbers of spanning trees through the counting of prime partitions

TL;DR

This paper investigates the set of integers representing the number of spanning trees in connected graphs with n vertices, showing that the set's size grows rapidly, thus answering a question posed by Sedlacek.

Contribution

It establishes a lower bound on the growth rate of the set of possible spanning tree counts, linking graph enumeration with prime partition counting.

Findings

|A_n| grows faster than sqrt{n}exp(2Pi*sqrt{n/log{n}/Sqrt(3)}

Provides a new asymptotic lower bound for the number of possible spanning tree counts

Answers a previously open question by Sedlacek about the growth of |A_n|

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. In this paper, we show that |A_n| grows faster than sqrt{n}exp(2Pi*sqrt{n/log{n}/Sqrt(3)} This settles a question of Sedlacek.