The Hopf Algebra of graph invariants

We propose an algebraic study of the simple graph isomorphism problem. We define a Hopf algebra from an explicit realization of its elements as formal power series. We show that these series can be evaluated on graphs and count occurrences of subgraphs. We establish a criterion for the isomorphism test of two simple graphs by means of occurrence counting of subgraphs. This criterion is deduced from algebraic relations between elements of our algebra.