On the second eigenvalue of random regular graphs

Expanders have many applications in Computer Science. It is known that random d-regular graphs are very efficient expanders, almost surely. However, checking whether a particular graph is a good expander is co-NP-complete. We show that the second eigenvalue of d-regular graphs, λ2, is concentrated in an interval of width O(√d) around its mean, and that its mean is O(d3/4). The result holds under various models for random d-regular graphs. As a consequence a random d-regular graph on n vertices, is, with high probability a certifiable efficient expander for n sufficiently large. The bound on the width of the interval is derived from martingale theory and the bound on E(λ2) is obtained by exploring the properties of random walks in random graphs.