  In this paper we investigate the connectedness and the isomorphism problems
for zig-zag products of two graphs. A sufficient condition for the zig-zag
product of two graphs to be connected is provided, reducing to the study of the
connectedness property of a new graph which depends only on the second factor
of the graph product. We show that, when the second factor is a cycle graph,
the study of the isomorphism problem for the zig-zag product is equivalent to
the study of the same problem for the associated pseudo-replacement graph. The
latter is defined in a natural way, by a construction generalizing the
classical replacement product, and its degree is smaller than the degree of the
zig-zag product graph.
  Two particular classes of products are studied in detail: the zig-zag product
of a complete graph with a cycle graph, and the zig-zag product of a
$4$-regular graph with the cycle graph of length $4$. Furthermore, an example
coming from the theory of Schreier graphs associated with the action of
self-similar groups is also considered: the graph products are completely
determined and their spectral analysis is developed.
