Let F(G) be the number of forests of a graph G. Similarly let C(G) be the number of connected spanning subgraphs of a connected graph G. We bound F(G) and C(G) for regular graphs and for graphs with a fixed average degree. Among many other things we study \(f_d=\sup _{G\in {\mathcal {G}}_d}F(G)^{1/v(G)}\), where \({\mathcal {G}}_d\) is the family of d-regular graphs, and v(G) denotes the number of vertices of a graph G. We show that \(f_3=2^{3/2}\), and if \((G_n)_n\) is a sequence of 3-regular graphs with the length of the shortest cycle tending to infinity, then \(\lim _{n\rightarrow \infty }F(G_n)^{1/v(G_n)}=2^{3/2}\). We also improve on the previous best bounds on \(f_d\) for \(4\le d\le 9\).

令F ( G ) 为图G的森林数量。类似地，设C ( G ) 是连通图G的连通生成子图的数量。我们为正则图和具有固定平均度的图绑定了F ( G ) 和C ( G )。在许多其他方面，我们研究\(f_d=\sup _{G\in {\mathcal {G}}_d}F(G)^{1/v(G)}\)，其中\({\mathcal {G }}_d\)是d -正则图族，v ( G ) 表示图G的顶点数. 我们证明了\(f_3=2^{3/2}\)，如果\((G_n)_n\)是一个 3-正则图序列，最短周期的长度趋于无穷大，那么\(\ lim _{n\rightarrow \infty }F(G_n)^{1/v(G_n)}=2^{3/2}\)。我们还改进了之前在\(f_d\) 上对\(4\le d\le 9\) 的最佳边界。