Expansion Of Product Replacement Graphs

We establish a connection between the expansion coefficient of the product replacement graph Γ _k (G) and the minimal expansion coefficient of a Cayley graph of G with k generators. In particular, we show that the product replacement graphs Γ _k (PSL(2,p)) form an expander family, under assumption that all Cayley graphs of PSL(2,p), with at most k generators are expanders. This gives a new explanation of the outstanding performance of the product replacement algorithm and supports the speculation that all product replacement graphs are expanders [42,52].