Expander Graph - Is the following graph family an expander graph?
Given $n$, consider the set of vertices $S = \{ (a,b,c) : c &lt; n/2 \} \subseteq V_n$. Assuming for simplicity that $n$ is even, this set contains exactly half the vertices. An edge connects $x \in S$ to $y \notin S$ iff $x = (a,b,n/2-1)$ and $y = (a,b,n/2)$, for a total of $n^2$ edges. Therefore the expansion of $S$ is $|E(S,\overline{S})|/|S| = n^2/(n^3/2) = 1/n$, which converges to zero.