  We consider three different models of sparse random graphs:~undirected and
directed Erd\H{o}s-R\'{e}nyi graphs, and random bipartite graph with an equal
number of left and right vertices. For such graphs we show that if the edge
connectivity probability $p \in (0,1)$ satisfies $n p \ge \log n + k(n)$ with
$k(n) \to \infty$ as $n \to \infty$, then the adjacency matrix is invertible
with probability approaching one (here $n$ is the number of vertices in the two
former cases and the number of left and right vertices in the latter case).
  If $np \le \log n -k(n)$ then these matrices are invertible with probability
approaching zero, as $n \to \infty$. In the intermediate region, when $np=\log
n + k(n)$, for a bounded sequence $k(n) \in \mathbb{R}$, the event $\Omega_0$
that the adjacency matrix has a zero row or a column and its complement both
have non-vanishing probability. For such choices of $p$ our results show that
conditioned on the event $\Omega_0^c$ the matrices are again invertible with
probability tending to one. This shows that the primary reason for the
non-invertibility of such matrices is the existence of a zero row or a column.
  The bounds on the probability of the invertibility of these matrices are a
consequence of quantitative lower bounds on their smallest singular values.
Combining this with an upper bound on the largest singular value of the
centered version of these matrices we show that the (modified) condition number
is $O(n^{1+o(1)})$ on the event that there is no zero row or column, with large
probability. This matches with von Neumann's prediction about the condition
number of random matrices up to a factor of $n^{o(1)}$, for the entire range of
$p$.
