Kazarian and Lando Seminars "Combinatorics of Vassiliev invariants" : S.K.Lando

The Hopf algebra of graphs, being a cocommutative graded connected Hopf algebra, is isomorphic to a polynomial Hopf algebra in its primitive elements.
An element p of a Hopf algebra with comultiplication Δ is said to be primitive if Δ(p)=1 x p+p x 1.
As a consequence, each space spanned by graphs with a given number of vertices decomposes into a direct sum of the subspace of primitive elements and the subspace spanned by disconnected graphs.

We will discuss a formula for the projection of the space spanned by the graphs to the subspace of primitive elements along the subspace spanned by disconnected graphs. Several explicit implementations of this formula for specific Hopf subgraphs will be given.