  Anosov diffeomorphisms are an important class of dynamical systems with many
peculiar properties. Ever since they were introduced in the sixties, it has
been an open question which manifolds can admit such diffeomorphisms, where
tori of dimension greater than or equal to two are the typical examples. It is
conjectured that the only manifolds supporting an Anosov diffeomorphism are
finitely covered by a nilmanifold, a type of manifold closely related to
rational nilpotent Lie algebras. In this paper, we study the existence of
Anosov diffeomorphisms for a large class of these nilpotent Lie algebras,
namely the ones that can be realized as a rational form in a Lie algebra
associated to a graph. From a given simple undirected graph, one can construct
a complex $c$-step nilpotent Lie algebra, which in general contains different
non-isomorphic rational forms, as described by the authors in previous work. We
determine precisely which forms correspond to a nilmanifold admitting an Anosov
diffeomorphism, leading to the first class of complex nilpotent Lie algebras
having several non-isomorphic rational forms and for which all the ones that
are Anosov are described. In doing so, we put a new perspective on certain
classifications in low dimensions and correct a false result in the literature.
