On the eigenvalues of Erdos-Renyi random bipartite graphs

Abstract

We analyse the eigenvalues of Erdös--Rényi random bipartite graphs. In particular, we consider $p$ satisfying $n_{1}p=\Omega (\sqrt{n_{1}p}{\log }^3(n_1)),$ $n_{2}p=\Omega (\sqrt{n_{2}p}{\log }^3(n_2)),$ and let $G\sim G(n_1,n_2,p)$. We show that with probability tending to $1$ as $n_1$ tends to infinity: $${\mu }_2(A(G))\le 2[1+o(1)](\sqrt{n_{1}p}+\sqrt{n_{2}p}+\sqrt{(n_1+n_2)p}).$$

5 pages; comments welcome