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.


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