arXiv 2408.013852024-08math.CO

KPKP图、孪生棒棒糖图及皮划桨图的正$e_I$-展开式

Positive $e_I$-expansions for KPKP graphs, twinned lollipops, and kayak paddles

Beijing Institute of Technology（北京理工大学）

Davion Q. B. Tang, David G. L. Wang

备注：: 23 pages, 8 figures; the positive e-expansion of kayak paddle graphs is corrected, an e-expansion of the infinity graph is added as a corollary

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AI总结：

该研究推导了KPKP图、孪生棒棒糖图及皮划桨图等图族的色对称函数的正$e_I$-展开式，完善了普通$e$-正性并补充了相关证明方法。

摘要：

我们推导了若干图族的色对称函数的显式正$e_I$-展开式。首先，得到KPKP图的统一公式，该公式可特化为棒棒糖图、KPK图、KKP图及PKP图的已知公式；接着，给出孪生路径与孪生环的正$e_I$-展开式，并将KPKP公式应用于孪生棒棒糖图；最后，建立皮划桨图的正$e_I$-展开式并将其特化为无穷图。这些公式通过保留组合索引系数，完善了普通$e$-正性，且提供了与现有递推、单位区间及非交换方法互补的独立证明。

英文摘要：

We derive explicit positive $e_I$-expansions for the chromatic symmetric functions of several graph families. First, we obtain a uniform formula for KPKP graphs that specializes to known formulas for lollipops, KPK graphs, KKP graphs, and PKP graphs. We then give positive $e_I$-expansions for twinned paths and cycles and apply the KPKP formula to twinned lollipops. Finally, we establish a positive $e_I$-expansion for kayak paddles and specialize it to infinity graphs. These formulas refine ordinary $e$-positivity by retaining composition-indexed coefficients and provide independent proofs complementary to existing recurrence, unit-interval, and noncommutative methods.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。