We prove that the chromatic symmetric function of any tree with a vertex of degree at least six is not e-positive, that is, it cannot be written as a nonnegative linear combination of elementary symmetric functions. This makes significant progress towards a recent conjecture of Dahlberg, She, and van Willigenburg, who conjectured the result for the chromatic symmetric functions of all trees with a vertex of degree at least four. We also provide a series of conditions that can identify when the chromatic symmetric function of a spider, which is a tree consisting of multiple paths all adjacent at a leaf to a center vertex, is not e-positive. These conditions generalize to trees and graphs with cut vertices as well. Finally, by using a result of Orellana and Scott, we give a method to inductively calculate certain coefficients in the elementary symmetric function expansion of the chromatic symmetric function of a spider, leading to further e-positivity conditions for spiders.

我们证明了任何具有至少六度顶点的树的色对称函数不是e-正的，也就是说，它不能写成初等对称函数的非负线性组合。这在 Dahlberg、She 和 van Willigenburg 最近的猜想方面取得了重大进展，他们猜想了顶点至少为四的所有树的色对称函数的结果。我们还提供了一系列条件，可以识别蜘蛛的色对称函数何时不是-积极的。这些条件也可以推广到具有切割顶点的树和图。最后，利用Orellana和Scott的结果，我们给出了一种方法来归纳计算蜘蛛色对称函数的初等对称函数展开中的某些系数，从而进一步得到蜘蛛的e-正性条件。