We consider three models of sparse random graphs: undirected and directed Erdős–Rényi graphs and random bipartite graph with two equal parts. For such graphs, we show that if the edge connectivity probability p satisfies \(np\ge \log n+k(n)\) with \(k(n)\rightarrow \infty \) as \(n\rightarrow \infty \), then the adjacency matrix is invertible with probability approaching one (n is the number of vertices in the two former cases and the same for each part in the latter case). For \(np\le \log n-k(n)\) these matrices are invertible with probability approaching zero, as \(n\rightarrow \infty \). 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 a 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. We further derive a bound on the (modified) condition number of these matrices on \(\Omega _0^c\), with a large probability, establishing von Neumann’s prediction about the condition number up to a factor of \(n^{o(1)}\).

我们考虑稀疏随机图的三种模型：无向和有向的Erdős-Rényi图以及具有相等的两部分的随机二分图。对于这样的图，我们表明如果边缘连通性概率p满足\（np \ ge \ log n + k（n）\）且\（k（n）\ rightarrow \ infty \）为\（n \ rightarrow \ infty \），则邻接矩阵是可逆的，概率接近1（n是前两种情况下的顶点数，而后一种情况下的每个部分都相同）。对于\（np \ le \ log nk（n）\）这些矩阵是可逆的，几率接近零，如\（n \ rightarrow \ infty \）。在中间区域，当\（np = \ log n + k（n）\），对于\ mathbb {R} \）中的有界序列\（k（n）\，事件\（\ Omega _0 \）邻接矩阵具有零行或一列并且其补码都没有消失可能性。对于p的这种选择，我们的结果表明，以事件\（\ Omega _0 ^ c \）为条件，矩阵再次可逆，且概率趋向于1。这表明此类矩阵不可逆的主要原因是存在零行或零列。我们进一步以很大的概率得出\（\ Omega _0 ^ c \）上这些矩阵的（修改的）条件数的界，从而建立了关于条件数的冯·诺依曼的预测，直到因子\（n ^ {o （1）} \）。