Two problems on independent sets in graphs

Let i t ( G ) denote the number of independent sets of size t in a graph G . Levit and Mandrescu have conjectured that for all bipartite G the sequence ( i t ( G ) ) t ≥ 0 (the independent set sequence of G ) is unimodal. We provide evidence for this conjecture by showing that this is true for almost all equibipartite graphs. Specifically, we consider the random equibipartite graph G ( n , n , p ) , and show that for any fixed p ∈ ( 0 , 1 ] its independent set sequence is almost surely unimodal, and moreover almost surely log-concave except perhaps for a vanishingly small initial segment of the sequence. We obtain similar results for p = Ω ̃ ( n − 1 / 2 ) . o consider the problem of estimating i ( G ) = ∑ t ≥ 0 i t ( G ) for G in various families. We give a sharp upper bound on the number of independent sets in an n -vertex graph with minimum degree δ , for all fixed δ and sufficiently large n . Specifically, we show that the maximum is achieved uniquely by K δ , n − δ , the complete bipartite graph with δ vertices in one partition class and n − δ in the other. o present a weighted generalization: for all fixed x > 0 and δ > 0 , as long as n = n ( x , δ ) is large enough, if G is a graph on n vertices with minimum degree δ then ∑ t ≥ 0 i t ( G ) x t ≤ ∑ t ≥ 0 i t ( K δ , n − δ ) x t with equality if and only if G = K δ , n − δ .