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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!

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2 Answers 2

up vote 13 down vote accepted

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.

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Just to show that this is an expander graph one just need a non-trivial bound towards Ramanujan's conjecture, so Sarnak-Xue result will be enough (as well as many others results). –  Asaf Mar 16 '13 at 22:18

Here is one secondary reference:

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


    p-Cycle


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Sorry for being unable to accept both answers... Thanks, anyway! –  Seva Mar 17 '13 at 7:38

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