Dense triangle-free $(n,d,\lambda )$-graphs for all orders

In 1994, Alon construct a triangle-free $(n,d,\lambda )$-graph with $d=\mathrm{\Omega }(n^{2\mathrm{/}3})$ and $\lambda =O(d^{1\mathrm{/}2})$ for an exponentially increasing sequence of integers $n$. Using his ingenious construction, we deduce that there exist triangle-free $(n,d,\lambda )$-graphs with $d=\mathrm{\Omega }(n^{2\mathrm{/}3})$ and $\lambda =O((d\log n{)}^{1\mathrm{/}2})$ for all sufficiently large $n$.