Abstract
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 Schmitt's more general formula for the antipode in an incidence Hopf algebra. Applications include several formulas (some old and some new) for evaluations of the Tutte polynomial. L'algèbre de graphes est une algèbre d'incidence de Hopf commutative, cocommutative, graduée, et connexe, dont les éléments de base correspondent à des graphes finis simples et dont le produit et coproduit de Hopf admettent une description combinatoire simple. Nous présentons une nouvelle formule de l'antipode dans l'algèbre de graphes utilisant les orientations acycliques; notre formule contient beaucoup moins de termes que la formule générale de Schmitt pour l'antipode dans une algèbre d'incidence de Hopf. Les applications incluent plusieurs formules (connues et inconnues) pour les évaluations du polynôme de Tutte.
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