We study the chromatic symmetric functions of graph classes related to spiders, namely generalized spider graphs (line graphs of spiders), and what we call horseshoe crab graphs. We show that no two generalized spiders have the same chromatic symmetric function, thereby extending the work of Martin, Morin and Wagner. Additionally, we establish Query Text="Please confirm if the inserted city name in affiliation [3] is correct. Amend if necessary." that a subclass of generalized spiders, which we call generalized nets, has no e-positive members, providing a more general counterexample to the necessity of the claw-free condition. We use yet another class of generalized spiders to construct a counterexample to a problem involving the e-positivity of claw-free, \(P_4\)-sparse graphs, showing that Tsujie’s result on the e-positivity of claw-free, \(P_4\)-free graphs cannot be extended to graphs in this set. Finally, we investigate the e-positivity of another type of graphs, the horseshoe crab graphs (a class of unit interval graphs), and prove the positivity of all but one of the coefficients. This has close connections to the work of Gebhard and Sagan and Cho and Huh.

我们研究与蜘蛛相关的图类的色对称函数，即广义蜘蛛图（蜘蛛的线图）以及所谓的马蹄蟹图。我们证明没有两个广义蜘蛛具有相同的色对称函数，从而扩展了Martin，Morin和Wagner的工作。此外，我们建立Query Text =“请确认隶属关系[3]中插入的城市名称是否正确。如有必要，请进行修改。” 广义蜘蛛的子类（我们称为广义网）没有e阳性成员，这为无爪条件的必要性提供了更一般的反例。我们使用另一类广义蜘蛛来构造一个涉及无爪\（P_4 \）的e阳性的问题的反例-稀疏图，表明Tsujie关于无爪\（P_4 \） -无图的e-阳性的结果不能扩展到该集中的图。最后，我们调查Ë另一种类型的图表，鲎图（一类单位间隔图形）的-positivity，并证明所有，但系数之一的积极性。这与格布哈德（Gebhard）和萨根（Sagan）以及乔和休（Hu and Huh）的工作紧密相关。