On the Second Eigenvalue of Random Bipartite Biregular Graphs

We consider the spectral gap of a uniformly chosen random ( d 1 , d 2 ) -biregular bipartite graph G with | V 1 | = n , | V 2 | = m , where d 1 , d 2 could possibly grow with n and m . Let A be the adjacency matrix of G . Under the assumption that d 1 ≥ d 2 and d 2 = O ( n 2 / 3 ) , we show that λ 2 ( A ) = O ( d 1 ) with high probability. As a corollary, combining the results from (Tikhomirov and Yousse in Ann Probab 47(1):362–419, 2019), we show that the second singular value of a uniform random d -regular digraph is O ( d ) for 1 ≤ d ≤ n / 2 with high probability. Assuming d 2 is fixed and d 1 = O ( n 2 ) , we further prove that for a random ( d 1 , d 2 ) -biregular bipartite graph, | λ i 2 ( A ) - d 1 | = O ( d 1 ) for all 2 ≤ i ≤ n + m - 1 with high probability. The proofs of the two results are based on the size biased coupling method introduced in Cook et al. (Ann Probab 46(1):72–125, 2018) for random d -regular graphs and several new switching operations we define for random bipartite biregular graphs.