An Expansion Tester for Bounded Degree Graphs

We consider the problem of testing graph expansion (either vertex or edge) in the bounded degree model (Goldreich & Ron, ECCC 2000). We give a property tester that takes as input a graph with degree bound d, an expansion bound α, and a parameter ε > 0. The tester accepts the graph with high probability if its expansion is more than α, and rejects it with high probability if it is ε-far from any graph with expansion α with degree bound d, where α < α is a function of α. For edge expansion, we obtain α = Ω( 2 d ), and for vertex expansion, we obtain α = Ω( 2 d2 ). In either case, the algorithm runs in time Õ( (1+μ)/2 d 2 εα2 ) for any fixed μ > 0.