Seminars "Combinatorics of invariants". Speaker: S.Lando

Many interesting graph invariants possess multiplicativity property, that is, the value of such an invariant on the disjoint union of graphs is the product of its values on the connected components. Hence, the disjoint union of graphs induces an algebra structure on the space of graphs.

What is less known, this space can be endowed with a second structure, namely, that of coalgebra: there is a natural way to define a coproduct of a graph. Once again, many interesting graph invariants behave nicely with respect to this second structure.

Moreover, the product and coproduct defined in this way are compatible which makes the space of graphs into a Hopf algebra. The same is true for spaces of chord diagrams modulo 4-term relations, delta-matroids, and many other combinatorial objects.

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