Expander properties and the cover time of random intersection graphs

We investigate important combinatorial and algorithmic properties of G n , m , p random intersection graphs. In particular, we prove that with high probability (a) random intersection graphs are expanders, (b) random walks on such graphs are “rapidly mixing” (in particular they mix in logarithmic time) and (c) the cover time of random walks on such graphs is optimal (i.e. it is Θ ( n log n ) ). All results are proved for p very close to the connectivity threshold and for the interesting, non-trivial range where random intersection graphs differ from classical G n , p random graphs.