Singularity of the k-core of a random graph

TL;DR

A key aspect of the 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.

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 neighbourhood. We prove that these kinds of dependencies are in some sense the only causes of singularity: for constants $k\ge 3$ and $λ> 0$, an Erdős--Rényi random graph $G\sim\mathbb{G}(n,λ/n)$ with $n$ vertices and edge probability $λ/n$ 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.

Theorems & Definitions (10)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Remark 1.5
• Remark 1.6
• Remark 1.7
• Lemma 3.1
• Theorem 3.2
• Lemma 3.3