The number of explicit constructions of expander graphs is still limited. I haven't kept up with the latest developments but I think this one is still open:

Find a 16-regular multigraph on $n$ vertices whose second largest eigenvalue $\lambda_2<8$.

Existence follows from a deep probabilistic result of Joel Friedman. It's possible that one can turn Friedman's proof into a randomized polynomial time algorithm because one can certainly test whether $\lambda_2<8$ in polynomial time, but I don't think there is a deterministic polynomial time algorithm known.