### A combinatorial non-commutative Hopf algebra of graphs

*Adrian Tanasa, Gerard H. E. Duchamp, Loic Foissy, Nguyen Hoang-Nghia, Dominique Manchon*

#### Abstract

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.

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