A note on the singularity probability of random directed d-regular graphs☆

In this note we show that the singular probability of the adjacency matrix of a random d-regular graph on n vertices, where d is fixed and n -> infinity, is bounded by n-1/3+o(1). This improves a recent bound by Huang in Huang (2021). Our method is based on the study of the singularity problem modulo a prime developed in Huang (2021) (and also partially in M & eacute;sz & aacute;ros, 2021; Nguyen and Wood, 2018), together with an inverse-type result on the decay of the characteristic function. The latter is related to the inverse Kneser's problem in combinatorics. (c) 2024 Published by Elsevier Ltd.