On Independent Sets in Graphs with Given Minimum Degree

The enumeration of independent sets in graphs with various restrictions has been a topic of much interest of late. Let i(G) be the number of independent sets in a graph G and let it(G) be the number of independent sets in G of size t. Kahn used entropy to show that if G is an r-regular bipartite graph with n vertices, then i(G) 6 i(Kr,r). Zhao used bipartite double covers to extend this bound to general r-regular graphs. Galvin proved that if G is a graph with δ(G) > δ and n large enough, then i(G) 6 i(Kδ,n−δ). In this paper, we prove that if G is a bipartite graph on n vertices with δ(G) > δ where n > 2δ, then it(G) 6 it(Kδ,n−δ) when t > 3. We note that this result cannot be extended to t = 2 (and is trivial for t = 0, 1). Also, we use Kahn’s entropy argument and Zhao’s extension to prove that if G is a graph with n vertices, δ(G) > δ, and ∆(G) 6 ∆, then i(G) 6 i(Kδ,∆).