Very sparse random graphs are known to typically be singular (i.e., have singular adjacency matrix), due to the presence of low-degree dependencies such as isolated vertices and pairs of degree-1 vertices with the same neighbourhood. We prove that these kinds of dependencies are in some sense the only causes of singularity: for constants $kge 3$ and $lambda > 0$, an ErdH os--Renyi random graph $Gsimmathbb{G}(n,lambda/n)$ with $n$ vertices and edge probability $lambda/n$ typically has the property that its $k$-core (its largest subgraph with minimum degree at least $k$) is nonsingular. This resolves a conjecture of Vu from the 2014 International Congress of Mathematicians, and adds to a short list of known nonsingularity theorems for extremely sparse random matrices with density $O(1/n)$. A key aspect of our proof is a technique to extract high-degree vertices and use them to boost the rank, starting from approximate rank bounds obtainable from (non-quantitative) spectral convergence machinery due to Bordenave, Lelarge and Salez.
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