  The graph algebra is a commutative, cocommutative, graded, connected
incidence Hopf algebra, whose basis elements correspond to finite simple graphs
and whose Hopf product and coproduct admit simple combinatorial descriptions.
We give a new formula for the antipode in the graph algebra in terms of acyclic
orientations; our formula contains many fewer terms than Takeuchi's and
Schmitt's more general formulas for the antipode in an incidence Hopf algebra.
Applications include several formulas (some old and some new) for evaluations
of the Tutte polynomial.
