# Expanders, Property Testing and the PCP theorem

## Lecture 5: Expanders, Probabilistic Proof of Existence

Posted in lectures by HQN on September 14, 2008

We gave a proof showing that most regular graphs are good (edge/vertex) expanders. In terms of edge-expansion, Bella Bollobas in this paper showed that, for every $d\geq 3$, most random $d$-regular graphs have edge-expansion rate (i.e. edge-isoperimetric number) at least $d/2 - c_1\sqrt d$ for some constant $c_1$. (Specifically, we can set $c_1=\sqrt{\lg 2}$. On the other hand, given any $d$-regular graph $G$ of order $n$, if we select a random subset $U$ of vertices of size $\lfloor n/2 \rfloor$, then the expected number of edges crossing the $(U,\bar U)$ cut is $\frac{\frac{dn}{2}\lfloor n/2 \rfloor \lceil n/2 \rceil}{\binom n 2}$

So the edge-isoperimetric number is bounded above by $\frac d 2 \frac{n+1}{n-1}$. Thus, for large graphs the edge expansion rate is (roughly) at most $\frac d 2$.

Better yet, Noga Alon proved in this paper that there is a constant $c_2$ such that the edge-isoperimetric number of any $d$-regular graph (with sufficiently large number of vertices) is at most $d/2 - c_2 \sqrt d$.

In summary, we can roughly think of edge-isoperimetric numbers of random $d$-regular graphs to be about $d/2 - \Theta(\sqrt d)$.

For small values of $d$, Bollobas’ analysis yields the following. For every sufficiently large $n$,

• there exists a $(n,3,2/11)$-edge expander
• there exists a $(n,4,11/25)$-edge expander
• there exists a $(n,5,18/25)$-edge expander
• there exists a $(n,6,26/25)$-edge expander
• there exists a $(n,9,2.06)$-edge expander