The cd-index is a noncommutative polynomial which compactly encodes the flag vector data of the face lattice of a polytope, and more generally, of an Eulerian poset. There is a simple yet powerful coalgebraic structure on the cd-index which enables one to understand how the cd-index of a polytope changes under geometric operations and proves non-trivial inequalities among the face incidence data.

We consider a general class of labeled graphs which satisfy a balanced condition and develop the cd-index. As a special case, this work applies to Bruhat graphs arising from the strong Bruhat order on a Coxeter group and gives straightforward proofs of recent results and conjectures of Billera and Brenti. I will also indicate various ongoing projects with Billera, Ehrenborg and Hetyei.

There are no prerequisites for this talk, other than knowing what "noncommutative" means.