Year: 2026
Type: preprint
Abstract: The theory of combinatorial Hopf algebras (CHA) is a powerful framework for studying algebraic invariants of combinatorial objects. For instance, Stanley's chromatic symmetric function and the chromatic polynomial arise naturally from the CHA of graphs. In this paper, we define a CHA of signed graphs in order to study, in the same spirit, invariants of signed graphs such as the signed chromatic polynomial. The key ingredient of our construction is a well-chosen character, from which the signed chromatic invariants are obtained. Via the antipode, we obtain new proofs of combinatorial reciprocity results for these invariants. Moreover, by deforming the character with an additional parameter, we incorporate bivariate invariants into this framework, both for classical and signed graphs. In the classical case, this technique recovers the Tutte polynomial.
Source: arXiv (Cornell University)
Authors: Aval, Jean-Christophe, Melgar, Raquel
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7208792643