On the Hopf algebra of multi-complexes

Abstract

The algebra of graphs is defined as the algebra which has a formal basis G of all isomorphism types of graphs, and multiplication is to take the disjoint union. We explicitly describe here the structure of the Hopf algebra of graphs H. We find an explicit basis B of the space of primitives, such that each graph is a polynomial with non-negative integer coefficients of the elements of B, and each b ∈ B is a polynomial with integer coefficients in G . Using this, we find the cancellation and grouping free formula for the antipode. The coefficients appearing in all these polynomials are, up to signs, numbers counting multiplicities of subgraphs in a graph. We then investigate applications of this to the graph reconstruction conjectures, and rederive some results in the literature on these questions.