On testing expansion in bounded-degree graphs

Authors:
Oded Goldreich;Dana Ron
Affiliations:
Faculty of Mathematics and Computer Science, The Weizmann Institute of Science, Israel;School of Electrical Engineering, Tel Aviv University, Israel
Venue:
Studies in complexity and cryptography
Year:
2011

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Abstract

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 0. The algorithm runs in time N0.5+α/poly(ε), and accepts any graph having (normalized) second eigenvalue at most λ. We believe that the algorithm rejects any graph that is ε-far from having second eigenvalue at most λα/O(1), and prove the validity of this belief under an appealing combinatorial conjecture.