I was wondering how to relate the spectra of the Zig-Zag product of two graphs in term of the factors...someone can help me?
The spectral gap is analyzed in the foundational paper 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.