Singularity of the k-core of a random graph

Abstract

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 neighborhood. We prove that these kinds of dependencies are in some sense the only causes of singularity: for constants $k\ge 3$ and $\mathit{\lambda }>0$, an Erdős–Rényi random graph $G\sim \mathbb{G}(n,\mathit{\lambda }∕n)$ with n vertices and edge probability $\mathit{\lambda }∕n$ 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 (nonquantitative) spectral convergence machinery due to Bordenave, Lelarge, and Salez.