  We generalize chain enumeration in graded partially ordered sets by relaxing
the graded, poset and Eulerian requirements. The resulting balanced digraphs,
which include the classical Eulerian posets having an $R$-labeling, imply the
existence of the (non-homogeneous) ${\bf cd}$-index, a key invariant for
studying inequalities for the flag vector of polytopes. Mirroring Alexander
duality for Eulerian posets, we show an analogue of Alexander duality for
balanced digraphs. For Bruhat graphs of Coxeter groups, an important family of
balanced graphs, our theory gives elementary proofs of the existence of the
complete ${\bf cd}$-index and its properties. We also introduce the rising and
falling quasisymmetric functions of a labeled acyclic digraph and show they are
Hopf algebra homomorphisms mapping balanced digraphs to the Stembridge peak
algebra. We conjecture nonnegativity of the ${\bf cd}$-index for acyclic
digraphs having a balanced linear edge labeling.
