An Expansion Tester for Bounded Degree Graphs

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We consider the problem of testing graph expansion (either vertex or edge) in the bounded degree model [10]. We give a property tester that given a graph with degree bound d, an expansion bound α, and a parameter ε> 0, 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 $\alpha' = \Omega(\frac{\alpha^2}{d})$ , and for vertex expansion, we obtain $\alpha' = \Omega(\frac{\alpha^2}{d^2})$ . In either case, the algorithm runs in time $\tilde{O}(\frac{n^{(1+\mu)/2}d^2}{\epsilon\alpha^2})$ for any given constant μ> 0.