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On the Number of Disconnected Character Degree Graphs Satisfying Pálfy's Inequality | ||
| International Journal of Group Theory | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 22 شهریور 1405 | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2026.145369.1964 | ||
| نویسندگان | ||
| Mark Lewis1؛ Andrew M Summers* 2 | ||
| 1Department of Mathematical Sciences, Kent State University, Kent, Ohio | ||
| 2Ph.D. candidate, Department of Mathematical Sciences, Kent State University, Kent, Ohio | ||
| چکیده | ||
| Let $G$ be a finite solvable group with disconnected character degree graph $\Delta(G)$. Under these conditions, it follows from a result of P. P. Pálfy that this graph consists of two connected components. Another result of P. P. Pálfy gives an inequality which relates the sizes of these two connected components (in terms of the number of vertices in each component). In this paper, we define the function $c(n)$ which calculates the number of possible component size pairs that satisfy Pálfy's inequality in terms of the order of $\Delta(G)$. 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. Finally, a table of examples is given to further illustrate the number of component size pairs which satisfy the inequality versus the total possible number of component size pairs when the order of the graph gets large. | ||
| کلیدواژهها | ||
| character degree graphs, Pálfy'؛ s inequality, finite solvable groups | ||
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