AI 中文总结
研究通过在特定路径顶点附加叶子得到的树，利用数论论证及相关结果，证明当n满足条件时其特征多项式不可约，从而得到无限多个具有不可约特征多项式的非同构树，解决了相关猜想。
AI 中文摘要
考虑通过在具有n - 1个顶点的路径的第三个顶点上附加一个叶子得到的树。在本笔记中，我们证明当n属于模30的某些等差数列时，该树的特征多项式是不可约的。因此，存在无限多个具有不可约特征多项式的两两非同构的树。我们的证明将几个数论论证与Gross、Hironaka和McMullen [GHM09]关于与图表En相关的Coxeter多项式的分圆因子的结果相结合。这肯定地解决了Akbari、Kumar、Mohar和Pragada的一个猜想。
英文摘要
Consider the tree obtained by attaching a leaf to the third vertex of a path with n-1 vertices. In this note, we prove that the characteristic polynomial of this tree is irreducible when n belongs to certain arithmetic progressions modulo 30. As a result there are infinitely many pairwise non-isomorphic trees with an irreducible the characteristic polynomial. Our proof combines several number-theoretic arguments with a result of Gross, Hironaka, and McMullen [GHM09] concerning the cyclotomic factors of the Coxeter polynomials associated with the diagrams En. This resolves affirmatively a conjecture of Akbari, Kumar, Mohar and Pragada.
Comments5 pages
