We show that almost surely the rank of the adjacency matrix of the Erdős‐Rényi random graph G ( n , p ) equals the number of nonisolated vertices for any c ln n / n ≤ p ≤ 1/2, where c is an arbitrary positive constant larger than 1/2. In particular, the adjacency matrix of the giant component (a.s.) has full rank in this range.
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Costello et al. (2008) studied this question.
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