Polyhedral joins and graph complexes

We give a decomposition of the suspension of a polyhedral join in terms of the polyhedral smash product of the suspension of the family of pairs, and study some cases in which the formula can be desuspended, particularly for polyhedral joins over independence complexes of graphs. We also give some lower bounds for the connectivity of polyhedral joins. We use these results to study the homotopy type of the forest filtration for some lexicographic products of graphs.