Invertibility of the 3-core of Erdos Renyi Graphs with Growing Degree

TL;DR

This paper proves that the adjacency matrix of the 3-core of Erdős–Rényi graphs with growing average degree is almost surely invertible as the number of nodes increases.

Contribution

It establishes the invertibility of the 3-core's adjacency matrix in Erdős–Rényi graphs with degrees growing to infinity but not exceeding logarithmic scale.

Findings

The 3-core's adjacency matrix is invertible with high probability.

Invertibility holds for degrees d growing to infinity but at most logarithmic in n.

Results apply to Erdős–Rényi graphs with specified degree conditions.

Abstract

Let $A \in \mathbb{R}^{n \times n}$ be the adjacency matrix of an Erd\H{o}s R\'enyi graph $G(n, d/n)$ for $d = \omega(1)$ and $d \leq 3\log(n)$. We show that as $n$ goes to infinity, with probability that goes to $1$, the adjacency matrix of the $3$-core of $G(n, d/n)$ is invertible.