Inequalities on the number of connected spanning subgraphs in a multigraph

Consider an undirected multigraph G = (V, E) with n vertices and m edges, and let N(i) denote the number of connected spanning subgraphs with i(m >= i >= n) edges in G. Recently, we showed in [3] the validity of (m - i + 1)N(i-1) > (i - n + [3+root 9 + 8(i-n)/2])N(i) for a simple graph and each i(m >= i >= n). Note that, from this inequality, (m-n)N(n)/2N(n+1) + N(n)/(m - n + 1)N(n-1) >= 2, is easily derived. In this paper, for a multigraph G and, all i(m >= i >= n), we prove (m - i + 1)N(i-1) >= (i - n + 2)N(i). In particular, this means that m - i + 1)N(i-1) > (i - n + [3+root 9 + 8(i-n)/2])N(i) is not valid for all multigraphs, in general. Furthermore, we prove (m-n)N(n)/2N(n+1) + N(n)/(m - n + 1)N(n-1) >= 2, which is not straightforwardly derived from (m - i + 1)N(i-1) >= (i - n + 2)N(i), and alsointroduce a necessary and sufficient condition by which (m - n)N(n)/2N(n+1) + N(n)/(m - n + 1)N(n-1) = 2. Moreover, we show a sufficient condition for a multigraph to have N(n)(2) > N(n-1)N(n+1). As special cases of the sufficient condition, we show that if G contains at least [2/3 (m - n)] + 1 multiple edges between some pair of vertices, or if its underlying simple graph has no cycle with length more than 4, then N(n)(2) > N(n-1)N(n+1).