Hopf algebras and factorial divergent power series: Algebraic tools for graphical enumeration

Abstract. In quantum field theory the number of primitive graphs, which are essentially multigraphs of higher edge-connectivity, is of special interest. The Connes-Kreimer Hopf algebra of graphs and the ring of factorially divergent power series can be used to extract complete asymptotic expansions for these graph classes. I will introduce both algebraic structures and give the complete asymptotic expansion of cyclically 4-edge-connected cubic graphs as an example. If time permits, I will briefly illustrate how these algebraic properties can be generalized to broader classes of graphs.