Generalized zig-zag products of regular digraphs and bounds on their spectral expansions

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.