  Anosov diffeomorphisms on closed Riemannian manifolds are a type of dynamical
systems exhibiting uniform hyperbolic behavior. Therefore their properties are
intensively studied, including which spaces allow such a diffeomorphism. It is
conjectured that any closed manifold admitting an Anosov diffeomorphism is
homeomorphic to an infra-nilmanifold, i.e. a compact quotient of a 1-connected
nilpotent Lie group by a discrete group of isometries. This conjecture
motivates the problem of describing which infra-nilmanifolds admit an Anosov
diffeomorphism.
  So far, most research was focused on the restricted class of nilmanifolds,
which are quotients of 1-connected nilpotent Lie groups by uniform lattices.
For example, Dani and Mainkar studied this question for the nilmanifolds
associated to graphs, which form the natural generalization of nilmanifolds
modeled on free nilpotent Lie groups. This paper further generalizes their work
to the full class of infra-nilmanifolds associated to graphs, leading to a
necessary and sufficient condition depending only on the induced action of the
holonomy group on the defining graph. As an application, we construct families
of infra-nilmanifolds with cyclic holonomy groups admitting an Anosov
diffeomorphism, starting from faithful actions of the holonomy group on simple
graphs.
