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, Erd\H{o}s, Malde and Schwenk made the conjecture that if $G$ is a tree then the independent set sequence $\{i_t(G)\}_{t\geq 0}$ of $G$ is unimodal; Levit and Mandrescu further conjectured that this should hold for all bipartite $G$.   We consider 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 $P(G,\lambda):=\sum_{t \geq 0} i_t(G)\lambda^t$ for these graphs, we obtain partial unimodality results in these cases.   We then 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 $t(d)/2^{d-1} > 1-1/\sqrt{2}$, 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 $Q_d$. 
David Galvin
math.CO