  We revisit the asymptotic analysis of probabilistic construction of adjacency
matrices of expander graphs proposed in [4]. With better bounds we derived a
new reduced sample complexity for the number of nonzeros per column of these
matrices, precisely $d = \mathcal{O}\left(\log_s(N/s) \right)$; as opposed to
the standard $d = \mathcal{O}\left(\log(N/s) \right)$. This gives insights into
why using small $d$ performed well in numerical experiments involving such
matrices. Furthermore, we derive quantitative sampling theorems for our
constructions which show our construction outperforming the existing
state-of-the-art. We also used our results to compare performance of sparse
recovery algorithms where these matrices are used for linear sketching.
