Lower bound on the nonzero Laplacian eigenvalue with the smallest real part

Consider a directed graph with $ n$ vertices. The graph is not assumed to be connected, and therefore the multiplicity of the eigenvalue 0 may be greater than 1. I am looking for a nonzero lower bound on the nonzero Laplacian eigenvalue with the smallest real part. The bound need not be very tight, but it must be a function of network size ($ n$ )