  We show that every $(n,d,\lambda)$-graph contains a Hamilton cycle for
sufficiently large $n$, assuming that $d\geq \log^{10}n$ and $\lambda\leq cd$,
where $c=\frac{1}{9000}$. This significantly improves a recent result of Glock,
Correia and Sudakov, who obtain a similar result for $d$ that grows
polynomially with $n$. The proof is based on the absorption technique combined
with a new result regarding the second largest eigenvalue of the adjacency
matrix of a subgraph induced by a random subset of vertices. We believe that
the latter result is of an independent interest and will have further
applications.
