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We show that every 
$(n,d,\lambda )$-graph contains a Hamilton cycle for sufficiently large 
$n$, assuming that 
$d\geq \log ^{6}n$ and 
$\lambda \leq cd$, where 
$c=\frac {1}{70000}$. This significantly improves a recent result of Glock, Correia, and Sudakov, who obtained a similar result for 
$d$ that grows polynomially with 
$n$. The proof is based on a new result regarding the second largest eigenvalue of the adjacency matrix of a subgraph induced by a random subset of vertices, combined with a recent result on connecting designated pairs of vertices by vertex-disjoint paths in 
$(n,d,\lambda )$-graphs. We believe that the former result is of independent interest and will have further applications.
