The spectral gap is analyzed in the [foundational paper][1] I don't know if you can say anything else.
Omer Reingold, Salil Vadhan, and Avi Wigderson, MR 1888797 Entropy waves, the zig-zag graph product, and new constant-degree expanders, Ann. of Math. (2) 155 (2002), no. 1, 157--187. [1]: http://www.math.ias.edu/~avi/PUBLICATIONS/MYPAPERS/SALIL/EXPANDERS/ANNALS/RVW01.pdf