The Maximum Number of Spanning Trees of a Graph with Given Matching Number

The number of spanning trees of a graph G is the total number of distinct spanning subgraphs of G that are trees. Feng et al. determined the maximum number of spanning trees in the class of connected graphs with n vertices and matching number β for 2 ≤ β ≤ n / 3 and β = ⌊ n / 2 ⌋ . They also pointed out that it is still an open problem to the case of n / 3 < β ≤ ⌊ n / 2 ⌋ - 1 . In this paper, we solve this problem completely.