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We consider three different models of sparse random graphs:~undirected anddirected Erd\H{o}s-R\'{e}nyi graphs, and random bipartite graph with an equalnumber of left and right vertices. For such graphs we show that if the edgeconnectivity 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 invertiblewith probability approaching one (here $n$ is the number of vertices in the twoformer 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 probabilityapproaching zero, as $n \to \infty$. In the intermediate region, when $np=\logn + k(n)$, 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 bothhave non-vanishing probability. For such choices of $p$ our results show thatconditioned on the event $\Omega_0^c$ the matrices are again invertible withprobability tending to one. This shows that the primary reason for thenon-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 aconsequence of quantitative lower bounds on their smallest singular values.Combining this with an upper bound on the largest singular value of thecentered version of these matrices we show that the (modified) condition numberis $O(n^{1+o(1)})$ on the event that there is no zero row or column, with largeprobability. This matches with von Neumann's prediction about the conditionnumber of random matrices up to a factor of $n^{o(1)}$, for the entire range of$p$.

Sharp transition of the invertibility of the adjacency matrices of sparse random graphs