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

@article{Basak2018SharpTO,
title={Sharp transition of the invertibility of the adjacency matrices of sparse random graphs},
author={Anirban Basak and Mark Rudelson},
journal={arXiv: Probability},
year={2018}
}
• Published 22 September 2018
• Mathematics
• arXiv: Probability
We consider three different models of sparse random graphs:~undirected and directed Erdős-Renyi 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…
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