标题： 关于满足帕尔菲不等式的非连通特征度图的个数 显示英文标题

标题： On the Number of Disconnected Character Degree Graphs Satisfying Pálfy's Inequality

摘要： 设$G$是一个具有不连通特征次数图$\Delta(G)$的有限可解群。在这种情况下，根据 Pálfy 的一个结果可知，$\Delta(G)$包含两个连通分支。Pálfy 的另一个结果给出了这两个连通分支大小之间的不等式关系。本文计算了满足 Pálfy 不等式的可能的分支大小对的数量。此外，对于固定的正整数$n$，证明了存在多少个不同的图阶数，使得恰好有$n$个分支大小对满足 Pálfy 的不等式。 显示英文摘要

摘要： Let $G$ be a finite solvable group with disconnected character degree graph $\Delta(G)$. Under these conditions, it follows from a result of P\'alfy that $\Delta(G)$ consists of two connected components. Another result of P\'alfy's gives an inequality relating the sizes of these two connected components. In this paper, we calculate the number of possible component size pairs that satisfy P\'alfy's inequality. Additionally, for a fixed positive integer $n$, the number of distinct graph orders for which exactly $n$ component size pairs satisfy P\'alfy's inequality is shown.