AI 中文总结
本文引入多轨道双变量色多项式，推导其元素级表示，给出概率解释，研究其不交并行为与无边图特化，为相关图论研究提供新工具。
AI 中文摘要
我们引入多轨道双变量色多项式$F_{\Gamma}(G;x,y)=\sum_{H\le G}\frac{1}{|H|}\sum_{h\in H}P_{\Gamma/h}(x,y)$，该多项式将作用于图$G$的有限群的子群格上的轨道双变量色多项式聚合起来。我们推导了等价的元素级表示$F_{\Gamma}(G;x,y)=\sum_{g\in G}c_G(g)P_{\Gamma/g}(x,y)$，其中$c_G(g)=\sum_{H\le G,\\,g\in H}\frac{1}{|H|}$。系数函数仅依赖于群元素生成的循环子群，且在共轭类上为常数。这产生了按循环子群和共轭类的相应分解，以及自然的莫比乌斯理论解释。归一化后，系数定义了作用群上的概率分布，使多轨道多项式可被解释为期望商多项式。我们进一步研究了其在不交并下的行为及其到无边图的特化，在后者中得到了加权循环指数表达式。
英文摘要
We introduce the multiorbital bivariate chromatic polynomial F_Γ(G;x,y) = \sum_{H\leq G}\frac{1}{|H|}\sum_{h\in H}P_{Γ/h}(x,y), which aggregates the orbital bivariate chromatic polynomials associated with all subgroups of a finite group acting on a graph. We derive the equivalent element-wise representation F_Γ(G;x,y) = \sum_{g\in G}c_G(g)P_{Γ/g}(x,y), where c_G(g) = \sum_{\langle g\rangle\leq H\leq G}\frac{1}{|H|}. The coefficient function depends only on the cyclic subgroup generated by the element and is constant on conjugacy classes. This yields decompositions by cyclic subgroups and conjugacy classes and an interpretation in terms of the incidence algebra of the subgroup lattice. After normalization, the coefficients define a probability distribution obtained by choosing a subgroup uniformly and then an element uniformly from that subgroup. We also establish diagonal multiplicativity for disjoint unions and a weighted cycle-index expression for edgeless graphs. A further contribution concerns the distinguishing power of the orbital bivariate chromatic polynomial. We answer a question of Dohmen and Lange-Geisler affirmatively by exhibiting two non-isomorphic graphs, P_3 and K_2 \mathbin{\dot\cup} K_1 under C_2-actions, with identical orbital bivariate chromatic polynomials. The two actions nevertheless have different multiorbital bivariate chromatic polynomials. Thus the multiorbital polynomial is not determined by the orbital bivariate polynomial, whereas the converse question remains open.
Comments26 pages
