We present a new approach to showing that random graphs are nearly optimal expanders. This approach is based on deep results from combinatorial group theory. It applies both to regular and irregular random graphs.
Let $G$ be a random d-regular graph on n vertices. It was conjectured by Alon (86') and proved by Friedman (08') in a $100\pm$page-long-booklet that the highest non-trivial eigenvalue of $G$ is a.a.s. arbitrarily close to $2\sqrt{(d-1)}$. We give a new, substantially simpler proof, that nearly recovers Friedman's result. This approach also has the advantage of applying to a more general model of random graphs, concerning also
non-regular graphs. Friedman (03') extended Alon's conjecture to this general case, and we obtain new, nearly optimal results here too.
