  Hilbert evolution algebras generalize evolution algebras through a framework
of Hilbert spaces. In this work we focus on infinite-dimensional Hilbert
evolution algebras and their representation through a suitably defined weighted
digraph. By means of studying such a digraph we obtain new properties for these
structures extending well-known results related to the nilpotency of finite
dimensional evolution algebras. We show that differently from what happens for
the finite dimensional evolution algebras, the notions of nil and nilpotency
are not equivalent for Hilbert evolution algebras. Furthermore, we exhibit
necessary and sufficient conditions under which a given Hilbert evolution
algebra is nil or nilpotent. Our approach includes illustrative examples.
