Well, if we limit our discussion to the weighted undirected graphs, there is Cheeger inequality which provides bounds for the 2nd eigenvalue in terms of other numerical characteristics of the graph: see e.g. here.
The result may still hold for weighted graphs directed graphs that are symmetrizable: namely, there exists a positive vector $\mu$ such that $A(x,y)\mu(x) = A(y,x)\mu(y)$, see in this very accessible lecture notes. Without this reversible/symmetrizable structure you don't have Green's theorem, so results are pretty weak.
As Tim has mentioned, though, there are no non-trivial bounds in general since the 2nd eigenvalue may be as close to the 1st one as one can imagine. So anyways, it all depends on a particular case. I don't know of any interesting estimates for the non-reversible case.