The combinatorial Hopf algebra of signed graphs

The theory of combinatorial Hopf algebras is a powerful framework for studying algebraic invariants of combinatorial objects. The aim of this work is to incorporate chromatic invariants of signed graphs into this context. We define the combinatorial Hopf algebra of signed graphs. Through this definition, algebraic invariants already known in the literature arise naturally. Using the antipode, we find new proofs of combinatorial reciprocity results for these invariants. We also study bivariate polynomial invariants, both for classical graphs and for signed graphs.