Q:

An expander (?) graph

For a prime $p$, consider the graph on the vertex set ${\mathbb F}_p$, in
which every vertex $z$ is adjacent to $z\pm 1$ and also to $z^{-1}$ (unless
$z=0$). I was told that this graph is known to be an expander, but the person
who told me this couldn't recall where exactly this graph has been studied.
Does anybody know the reference? Thanks!

A:

Here is Theorem 4.4.2 in Lubotzky's lovely book "Discrete groups, expanding graphs and invariant measures". On the projective line over $\mathbb{F}_p$, connect $z$ to $z\pm 1$ and to $-\frac{1}{z}$; this is a family of 3-regular expander graphs. The proof is indeed based on Selberg's 3/16-theorem.

A:

Here is one secondary reference:

Salil Vadhan, Lecture Notes, 2009.  Chapter 4: Expander Graphs. p.62.
  (PDF download link)

   

