Formulas for Counting the Numbers of Connected Spanning Subgraphs with at Most n+1 Edges in a Complete Graph K_n

Abstract

Let N_i be the number of connected spanning subgraphs with i(n-1≤i≤m) edges in an n-vertex m-edge undirected graph G=(V,E). Although N_n-1 is computed in polynomial time by the Matrixtree theorem, whether N_n is efficiently computed for a graph G is an open problem (see e. g., [2]). On the other hand, whether N²_n≥N_n-1N_n+1 for a graph G is also open as a part of log concave conjecture (see e. g., [6], [12]). In this paper, for a complete graph K_n, we give the formulas for N_n, N_n+1, by which N_n, N_n+1 are respectively computed in polvnomial time on n, and, in particular, prove N²_n>(n-1)(n-2)/n(n-3)N_n-1N_n+1 as well.