  Let $G$ be a finite group and let $H_1,H_2<G$ be two subgroups. In this
paper, we are concerned with the bipartite graph whose vertices are $G/H_1\cup
G/H_2$ and a coset $g_1H_1$ is connected with another coset $g_2H_2$ if and
only if $g_1H_1\cap g_2 H_2\neq\varnothing$. The main result of the paper
establishes the existence of such graphs with large girth and large spectral
gap. Lubotzky, Manning and Wilton use such graphs to construct certain infinite
groups of interest in geometric group theory.
