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

• Authors: Peng Cheng;Shigeru Masuyama
• Venue: IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
• Year: 2008

Let Ni be the number of connected spanning subgraphs with i(n-1 im) edges in an n-vertex m-edge undirected graph G = (V,E). Although Nn-1 is computed in polynomial time by the Matrix-tree theorem, whether Nn is efficiently computed for a graph G is an open problem (see e.g., [2]). On the other hand, whether Nn2 ≥ Nn-1Nn+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 Kn, we give the formulas for Nn, Nn+1, by which Nn, Nn+1 are respectively computed in polynomial time on n, and, in particular, prove Nn2 (n-1)(n-2)/n(n-3) Nn-1Nn+1 as well.