  Given a directed graph, an equivalence relation on the graph vertex set is
said to be balanced if, for every two vertices in the same equivalence class,
the number of directed edges from vertices of each equivalence class directed
to each of the two vertices is the same. In this paper we describe the quotient
and lift graphs of symmetric directed graphs associated with balanced
equivalence relations on the associated vertex sets. In particular, we
characterize the quotients and lifts which are also symmetric. We end with an
application of these results to gradient and Hamiltonian coupled cell systems,
in the context of the coupled cell network formalism of Golubitsky, Stewart and
Torok(Patterns of synchrony in coupled cell networks with multiple arrows.
{SIAM Journal of Applied Dynamical Systems, 4 (1) (2005) 78-100).
