Consider an undirected multigraph G=(V,E) with n vertices and m edges, and let Ni denote the number of connected spanning subgraphs with i(min) edges in G. Recently, we showed in [3] the validity of (m-i+1)Ni-1>Ni for a simple graph and each i(min). Note that, from this inequality, 2 is easily derived. In this paper, for a multigraph G and all i(min), we prove (m-i+1)Ni-1(i-n+2)Ni, and give a necessary and sufficient condition by which (m-i+1)Ni-1=(i-n+2)Ni. In particular, this means that (m-i+1)Ni-1>Ni is not valid for all multigraphs, in general. Furthermore, we prove 2, which is not straightforwardly derived from (m-i+1)Ni-1(i-n+2)Ni, and also introduce a necessary and sufficent condition by which =2. Moreover, we show a sufficient condition for a multigraph to have Nn2>Nn-1Nn+1. As special cases of the sufficient condition, we show that if G contains at least +1 multiple edges between some pair of vertices, or if its underlying simple graph has no cycle with length more than 4, then Nn2>Nn-1Nn+1.

Consider an undirected graph G=(V,E) with n (=|V|) vertices and m (=|E|) edges. It is well-known that the problem of computing the sequence Nn-1,Nn,...,Nm is #P-complete (see e.g.,[3]), where Ni denotes the number of connected spanning subgraphs with i (n-1!im) edges in G. In this paper, by proving new inequalities on the sequence Nn-1,Nn,...,Nm, we show an interesting and stronger property that the sequence γn-1,γn,...,γm, where γi denotes the average number of spanning trees in the connected spanning subgraphs with i edges, is a convex sequence as well as a monotonically increasing sequence, although this property does not hold for the sequence Nn-1,Nn,...,Nm.