  We introduce a generalization of the zig-zag product of regular digraphs
(directed graphs), which allows us to construct regular digraphs with m ore
flexible choices of the degrees. In our generalization, we can control the
connectivity of the resulting graph measured by its spectral expansion. We
derive an upper bound on the spectral expansion of the generalized zig-zag
product. Our upper bound improves on known bounds when applied to the zig-zag
product. We also consider a special case of the generalized zig-zag product,
where one of the components is a trivial graph whose edges are all self-loops.
We call it a reduced zig-zag product and derive a bound on the spectral
expansion of its powers.
