Margolis and Meakin use the Cayley graph of a group presentation to constructE-unitary inverse monoids [11]. This is the technique we refer to asgraph expansion. In this paper we consider graph expansions ofunipotentmonoids, where a monoid is unipotent if it contains a unique idempotent. The monoids arising in this way areE-unitary and belong to the quasivariety ofweakly left amplemonoids. We give a number of examples of such monoids. We show that the least unipotent congruence on a weakly left ample monoid is given by the same formula as that for the least group congruence on an inverse monoid and we investigate the notion ofproperfor weakly left ample monoids.