AI 中文总结
本研究定义了带符号图的组合Hopf代数，利用其对极给出相关不变量组合互易结果的新证明，还研究了两类图的双变量多项式不变量，拓展了组合Hopf代数在带符号图领域的应用。
AI 中文摘要
组合Hopf代数理论是研究组合对象代数不变量的强大框架。本研究旨在将带符号图的色不变量纳入该框架，定义了带符号图的组合Hopf代数，通过该定义自然得到了文献中已有的代数不变量；利用对极（antipode），为这些不变量的组合互易结果提供了新的证明；还研究了经典图与带符号图的双变量多项式不变量。
英文摘要
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.
Comments21 pages, 4 figures
