Simple Constructions of Special Graph Families
This is a family of expander graphs: let $p$ be a prime, let the vertex set be $\{ 0,\ldots,p-1\}$, connect vertex $x$ to $x+ 1 \bmod p$, to $x-1 \bmod p$ and to $x^{-1} \bmod p$. Each such graph is a cycle, plus a matching over $p-3$ vertices. (The multiplicative inverse is undefined for $0$, and is the element itself for $p-1$ and $1$.)

This is a picture of what it looks like for $p=59$



There is no known elementary proof that this a family of expanders, but the construction is very simple.