On the Hopf algebra of graphs

Miodrag Iovanov, Jaiung Jun

Published 2019-12-23Version 1

The algebra of graphs is defined as the algebra which has a formal basis $\mathcal{G}$ 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 $\mathcal{B}$ of the space of primitives, such that each graph is a polynomial with non-negative integer coefficients of the elements of $\mathcal{B}$, and each $b\in\mathcal{B}$ is a polynomial with integer coefficients in $\mathcal{G}$. 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.