AI 中文总结
本文研究可解群特征标度图的谱算术性质，推导(n-2)-正则图的特征多项式公式，分析超图、Lewis图及直径3图的分裂域与特征值相关性质。
AI 中文摘要
本文研究特征标度图的特征值，尤其关注其谱的算术性质。首先，研究可解群的(n-2)-正则特征标度图，推导其特征多项式的显式公式，证明其所有特征值均为有理数，因此其分裂域为ℚ。接着，考虑在这些图上添加边得到的超图，证明对应的分裂域是ℚ的二次扩域。然后，利用结构分解，研究一类通用的Lewis图，针对该类图，建立无理特征值数量及相关分裂域次数的界。最后，研究直径为3的素特征标度图，重点关注其特征值的算术性质及分裂域的次数。
英文摘要
In this paper, we investigate the eigenvalues of character degree graphs, with particular emphasis on the arithmetic properties of their spectra. First, we study \((n-2)\)-regular character degree graphs of solvable groups and derive an explicit formula for their characteristic polynomials. We show that all their eigenvalues are rational and, consequently, that their splitting field is \(\mathbb{Q}\). We then consider supergraphs obtained by adding edges to these graphs and prove that the corresponding splitting field is a quadratic extension of \(\mathbb{Q}\). Next, using their structural decomposition, we examine a general class of Lewis graphs. For this class, we establish bounds on both the number of irrational eigenvalues and the degree of the associated splitting fields. Finally, we investigate prime character degree graphs of diameter \(3\), focusing on the arithmetic nature of their eigenvalues and the degree of their splitting fields.
Comments24 pages, 5 figures, comments welcome
