Authors: Adrian Tanasa ^1,2; Gerard Duchamp ¹; Loïc Foissy ³; Nguyen Hoang-Nghia ¹; Dominique Manchon ⁴

• 1 Laboratoire d'Informatique de Paris-Nord
• 2 Horia Hulubei National Institute for Physics and Nuclear Engineering
• 3 Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville
• 4 Laboratoire de Mathématiques Blaise Pascal

A non-commutative, planar, Hopf algebra of planar rooted trees was defined independently by one of the authors in Foissy (2002) and by R. Holtkamp in Holtkamp (2003). In this paper we propose such a non-commutative Hopf algebra for graphs. In order to define a non-commutative product we use a quantum field theoretical (QFT) idea, namely the one of introducing discrete scales on each edge of the graph (which, within the QFT framework, corresponds to energy scales of the associated propagators). Finally, we analyze the associated quadri-coalgebra and codendrifrom structures.