On the Minimal Graph with a Given Number of Spanning Trees

Abstract

Let G be a finite connected graph without loops or multiple edges. A maximal tree subgraph T of G is called a spanning tree of G. Denote by k(G) the number of all trees spanning the graph G. A. Rosa formulated the following problem (private communication): Let x(≠2) be a given positive integer and denote by α(x) the smallest positive integer y having the following property: There exists a graph G on y vertices with x spanning trees. Investigate the behavior of the function α(x).

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

“…these are the only numbers n such that α(n) = n. He also defined the function β(n) as the least number of edges l for which there exists a graph with l edges and with precisely n spanning trees. He showed that α(n) < β(n) n + 1 2 , except for the fixed points of α in which case it holds that α(n) = β(n) = n. Moreover, as is observed in [8], from the construction used by Sedláček [7] we have…”

“…We use this number theoretical result to improve the answer related to the question Sedláček [7] posed in 1970: Given a number n 3, what is the least number k such that there exists a graph on k vertices having precisely n spanning trees? Sedláček denoted this function by α(n).…”

On the minimum number of vertices and edges in a graph with a given number of spanning trees

“…Obviously a(x) ^ fi(x) ^ x, for any x ^ 3. The function a has been studied by J. SEDLACEK [3], who also gave an impulse to the rise of the present paper.…”

“…The very simple generalization of one of the procedures used in [3] for the estimate of the function a leads to the following estimate of the function p which is given by graphs with at least one separating vertex: if x x and x 2 are integers and…”

“…By making use of the graph D(l, 2, (x -2)/3) and a graph with no separating edge and with two circuits of length 3 and x/3, J. Sedlacek [3] found an upper estimate of the function a for almost all x = 2, 3 (mod 3). By using the same graphs it is quite readily possible to find an estimate of the function p for the same values of the argument:…”