On Testing Expansion in Bounded-Degree Graphs

We consider testing graph expansion in the bounded-degree graph model. Specifically, we refer to algorithms for testing whether the graph has a second eigenvalue bounded above by a given threshold or is far from any graph with such (or related) property. We present a natural algorithm aimed towards achieving the foregoing task. The algorithm is given a (normalized) eigenvalue bound λ < 1, oracle access to a bounded-degree N-vertex graph, and two additional parameters ǫ, α > 0. The algorithm runs in time N/poly(ǫ), and accepts any graph having (normalized) second eigenvalue at most λ. We believe that the algorithm rejects any graph that is ǫ-far from having second eigenvalue at most λ, and prove the validity of this belief under an appealing combinatorial conjecture.