On the Number of Forests and Connected Spanning Subgraphs
AbstractLet 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 $$f_d=\sup _{G\in {\mathcal {G}}_d}F(G)^{1/v(G)}$$ f d = sup G ∈ G d F ( G ) 1 / v ( G ) , where $${\mathcal {G}}_d$$ G d is the family of d-regular graphs, and v(G) denotes the number of vertices of a graph G. We show that $$f_3=2^{3/2}$$ f 3 = 2 3 / 2 , and if $$(G_n)_n$$ ( G n ) n is a sequence of 3-regular graphs with the length of the shortest cycle tending to infinity, then $$\lim _{n\rightarrow \infty }F(G_n)^{1/v(G_n)}=2^{3/2}$$ lim n → ∞ F ( G n ) 1 / v ( G n ) = 2 3 / 2 . We also improve on the previous best bounds on $$f_d$$ f d for $$4\le d\le 9$$ 4 ≤ d ≤ 9 .