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The Rank of Random Graphs

Kevin P. Costello·Van H. Vu
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Abstract

We show that almost surely the rank of the adjacency matrix of the Erdös-Rényi random graph $G(n,p)$ equals the number of non-isolated vertices for any $c\ln n/n<p<1/2$, where $c$ is an arbitrary positive constant larger than 1/2. In particular, the giant component (a.s.) has full rank in this range.

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Kevin P. Costello, Van H. Vu. 2006-06-17. The Rank of Random Graphs. https://arxiv.org/abs/math/0606414

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