On the Second Eigenvalue of Random Bipartite Biregular Graphs

Abstract

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 $$d_1\ge d_2$$ and $$d_2=O(n^{2/3}),$$ we show that $$\lambda _2(A)=O(\sqrt{d_1})$$ 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 $$O(\sqrt{d})$$ for $$1\le d\le n/2$$ 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, $$|\lambda _i^2(A)-d_1|=O(\sqrt{d_1})$$ for all $$2\le i\le n+m-1$$ 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.