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

Let $G$ be a finite solvable group with disconnected character degree graph $\mathrm{\Delta }(G)$. Under these conditions, it follows from a result of Pálfy that $\mathrm{\Delta }(G)$ consists of two connected components. Another result of Pálfy’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 Pálfy’s inequality. Additionally, for a fixed positive integer $n$, the number of distinct graph orders for which exactly $n$ component size pairs satisfy Pálfy’s inequality is shown.