On the rank, Kernel, and core of sparse random graphs

Abstract

AbstractWe study the rank of the adjacency matrix of a random Erdős‐Rényi graph . It is well known that when , with high probability, is singular. We prove that when , with high probability, the corank of is equal to the number of isolated vertices remaining in after the Karp‐Sipser leaf‐removal process, which removes vertices of degree one and their unique neighbor. We prove a similar result for the random matrix , where all entries are independent Bernoulli random variables with parameter . Namely, we show that if is the bipartite graph with bi‐adjacency matrix , then the corank of is with high probability equal to the max of the number of left isolated vertices and the number of right isolated vertices remaining after the Karp‐Sipser leaf‐removal process on . Additionally, we show that with high probability, the ‐core of is full rank for any and . This partially resolves a conjecture of Van Vu for . Finally, we give an application of the techniques in this paper to gradient coding, a problem in distributed computing.