Lemma 4.7 (Expander Mixing Lemma). Assuming that:

$G = (V, E)$ an $(n, d, λ)$ -graph
$S, T \subseteq V$ (and define $e (S, T) = \sum_{x \in S} \sum_{y \in T} 𝟙_{x y \in E}$ )

Then

\begin{array}{l} | e (S, T) - \frac{d}{n} | S | | T | | & \leq \frac{λ}{n} \sqrt{| S | | S^{c} | | T | | T^{c} |} \\ \leq λ \sqrt{| S | | T |} \end{array}