We prove an analogue of Alon's spectral gap conjecture for random bipartite, biregular
graphs. We use the Ihara–Bass formula to connect the non-backtracking spectrum to that of …

Let D n, d be the set of all d-regular directed graphs on n vertices. Let G be a graph chosen
uniformly at random from D n, d and M be its adjacency matrix. We show that M is invertible …

Let M n be an n× n random matrix with independent and identically distributed Bernoulli (p)
entries. We show that there is a universal constant C≥ 1 such that, whenever p and n satisfy …

Fix a constant C≥ 1 and let d= d (n) satisfy d≤ lnC n for every large integer n. Denote by An
the adjacency matrix of a uniform random directed d-regular graph on n vertices. We show …

We derive a lower bound on the smallest singular value of a random d-regular matrix, that is,
the adjacency matrix of a random d-regular directed graph. Specifically, let C_1< d< cn/\log …

Let $ d $ and $ n $ be integers satisfying $ C\leq d\leq\exp (c\sqrt {\ln n}) $ for some
universal constants $ c, C> 0$, and let $ z\in\mathbb {C} $. Denote by $ M $ the adjacency …

Let $ A_n $ be an $ n\times n $ random matrix with iid Bernoulli ($ p $) entries. For a fixed
positive integer $\beta $, suppose $ p $ satisfies $$\frac {\log (n)}{n}\le p\le c_\beta $$ where …

Let d be a (large) integer. Given n≥ 2 d, let A n be the adjacency matrix of a random
directed d-regular graph on n vertices, with the uniform distribution. We show that the rank of …