  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.
