6 research outputs found
A combinatorial non-commutative Hopf algebra of graphs
CombinatoricsInternational audienceA 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
Primitive elements of a connected free bialgebra
We prove that the Lie algebra of primitive elements of a graded and connected
bialgebra, free as an associative algebra, over a eld of characteristic zero,
is a free Lie algebra. The main tool is a ltration, which allows to embed the
associated graded Lie algebra into the Lie algebra of a free and cocommutative
bialgebra. The result is then a consequence of Cartier-Quillen-Milnor-Moore's
Shirshov-Witt's theorems
Recipe theorems for polynomial invariants on ribbon graphs with half-edges
We provide recipe theorems for the Bollob\`as and Riordan polynomial
defined on classes of ribbon graphs with half-edges introduced in
arXiv:1310.3708[math.GT]. We also define a generalized transition polynomial
on this new category of ribbon graphs and establish a relationship between
and .Comment: 24 pages, 14 figure