The independent set sequence of regular bipartite graphs

Let i t ( G ) be the number of independent sets of size t in a graph G . Alavi, Erdős, Malde and Schwenk made the conjecture that if G is a tree then the independent set sequence { i t ( G ) } t ≥ 0 of G is unimodal; Levit and Mandrescu further conjectured that this should hold for all bipartite G . sider the independent set sequence of finite regular bipartite graphs, and graphs obtained from these by percolation (independent deletion of edges). Using bounds on the independent set polynomial P ( G , λ ) ≔ ∑ t ≥ 0 i t ( G ) λ t for these graphs, we obtain partial unimodality results in these cases. n focus on the discrete hypercube Q d , the graph on vertex set { 0 , 1 } d with two strings adjacent if they differ on exactly one coordinate. We obtain asymptotically tight estimates for i t ( d ) ( Q d ) in the range t ( d ) / 2 d − 1 > 1 − 1 / 2 , and nearly matching upper and lower bounds otherwise. We use these estimates to obtain a stronger partial unimodality result for the independent set sequence of Q d .