  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 $k\ge 3$ and $\lambda > 0$, an
Erd\H{o}s--R\'enyi random graph $G\sim\mathbb{G}(n,\lambda/n)$ 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.
