Abstract
Let d≥3 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 d>0 such that P(A is singular in R)≤n−d, 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 Fp. The proof combines a local central limit theorem and a large deviation estimate.
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