Let $\Gamma \, zz\, G$ denote the zig-zag product of graphs $\Gamma$ and $G$. Reingold, Vadhan, and Wigderson proved that the second largest eigenvalue of the normalized adjacency matrix of the zig-zag product satisfies $$ \lambda(\Gamma\, zz\, G)\leq \lambda(\Gamma) + \lambda(G). $$ This leads to a simple construction of expanders: given a graph $\Gamma$ with sufficiently small $\lambda(\Gamma)$, the sequence $$ \Gamma_n=\Gamma_{n-1}^2 \, zz\, \Gamma $$ is an expanding family, where $\Gamma^2$ is the square of the graph. Basically, the zig-zag product reduces the degree, while the squaring reduces the second eigenvalue.

**Question:** Is it possible to get expanders using just the iterated zig-zag product without squaring, i.e., is there a graph $\Gamma$ such that $\Gamma_n=\Gamma_{n-1} \, zz\, \Gamma$ is an expanding family?
Or the zig-zag product necessary increases the second eigenvalue?