Some graph products and their expansion properties

We introduce "derandomized" versions of the tensor product and the zig-zag product, extending the ideas in the derandomized squaring operation of Rozenman and Vadhan. These enable us to obtain graphs with smaller degrees than those obtained using their non-derandomized counterparts, though at the cost of slightly worse expansion. In this paper we give bounds on these expansions (measured by their second eigenvalues), and also obtain an improved bound on the expansion of the derandomized square.