Let be a fixed integer, and let A be the adjacency matrix of a random d-regular directed or undirected graph on n vertices. We show that there exists a constant such that
for n sufficiently large. This answers an open problem by Frieze and Vu. The key idea is to study the singularity probability over a finite field . The proof combines a local central limit theorem and a large deviation estimate.
"Invertibility of adjacency matrices for random d-regular graphs." Duke Math. J. 170 (18) 3977 - 4032, 1 December 2021. https://doi.org/10.1215/00127094-2021-0006