  It was conjectured by Alon and proved by Friedman that a random $d$-regular
graph has nearly the largest possible spectral gap, more precisely, the largest
absolute value of the non-trivial eigenvalues of its adjacency matrix is at
most $2\sqrt{d-1} +o(1)$ with probability tending to one as the size of the
graph tends to infinity. We give a new proof of this statement. We also study
related questions on random $n$-lifts of graphs and improve a recent result by
Friedman and Kohler.
