The combinatorial Hopf algebra of signed graphs
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.
Related
- Hopf Algebra of Graphs for Operator Splitting Methods
- An upper bound on the smallest singular value of dense signed random combinatorial matrices
- Hopf algebra polynomial identities in the Sweedler algebra
- Hopf Images of Hopf algebra Coactions
- The Quillen and sharbly Hopf algebra structures on Steinberg homology coincide
BibTeX
@misc{aval2026combinatorial,
title = {{The combinatorial Hopf algebra of signed graphs}},
author = {Jean-Christophe Aval and Raquel Melgar},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.03959},
eprint = {2609.03959},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03959},
}
RIS
TY - UNPB
TI - The combinatorial Hopf algebra of signed graphs
AU - Jean-Christophe Aval
AU - Raquel Melgar
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2609.03959
UR - https://arxiv.org/abs/2609.03959
AB - 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.
ER -