Definition 3.3 (Edge expansion of a graph). The edge expansion of a graph is

Φ (G) : = \min_{\begin{array}{c} S \subseteq V \\ \emptyset \neq S \neq V \end{array}} Φ (S, V ∖ S) = \min_{\begin{array}{c} S \subseteq V \\ 0 < | S | < | V | ∕ 2 \end{array}} Φ (S) .