A Hopf-algebraic structure for generalized transition polynomials.
In 1987 Jaeger defined transition polynomials on four-regular graphs. These provide a common framework to study polynomials given by linear recursion relations based on transitions (vertex states) rather than deletion-contraction. We generalize transition polynomials to arbitrary Eulerian graphs and provide a Hopf algebra such that these polynomials are Hopf maps. The resulting comultiplcation and antipode then lead to new results for these polynomials, which include the Penrose, Martin and Tutte polynomials.