Testing the expansion of a graph

Abstract

We study the problem of testing the expansion of graphs with bounded degree d in sublinear time. A graph is said to be an $\alpha$-expander if every vertex set $U\subset V$ of size at most $\frac{1}{2}|V|$ has a neighborhood of size at least $\alpha |U|$.

We show that the algorithm proposed by Goldreich and Ron [9] (ECCC-2000) for testing the expansion of a graph distinguishes with high probability between $\alpha$-expanders of degree bound d and graphs which are $$-far from having expansion at least $\mathrm{\Omega }({\alpha }^2)$. This improves a recent result of Czumaj and Sohler [3] (FOCS-07) who showed that this algorithm can distinguish between $\alpha$-expanders of degree bound d and graphs which are $$-far from having expansion at least $\mathrm{\Omega }({\alpha }^2/\log n)$. It also improves a recent result of Kale and Seshadhri [12] (ECCC-2007) who showed that this algorithm can distinguish between $\alpha$-expanders and graphs which are $$-far from having expansion at least $\mathrm{\Omega }({\alpha }^2)$ with twice the maximum degree. Our methods combine the techniques of [3], [9] and [12].