On the number of disconnected character degree graphs satisfying Palfy’s inequality

Lewis, Mark L.; Summers, Andrew

doi:10.22108/ijgt.2026.145369.1964

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	On the number of disconnected character degree graphs satisfying Palfy’s inequality
International Journal of Group Theory
مقالات آماده انتشار، اصلاح شده برای چاپ، انتشار آنلاین از تاریخ 07 مهر 1405 اصل مقاله (365.31 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22108/ijgt.2026.145369.1964
نویسندگان
Mark L. Lewis؛ Andrew Summers^*
Department of Mathematical Sciences, Kent State University, Kent OH 44242, USA
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. P'alfy’s 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؛ P\'alfy's inequality؛ finite solvable groups

مراجع
[1] M. W. Bissler, J. Laubacher and M. L. Lewis, Classifying character degree graphs with six vertices, Beitr. Algebra Geom., 60 no. 3 (2019) 499–511. [2] I. M. Isaacs, Character theory of finite groups, Dover Publications Inc., New York, 1994. [3] M. L. Lewis, An overview of graphs associated with character degrees and conjugacy class sizes in finite groups, Rocky Mountain J. Math., 38 no. 1 (2008) 175–211. [4] M. L. Lewis, Classifying character degree graphs with 5 vertices, Finite groups 2003, Walter de Gruyter, Berlin, 2004, 247–265. [5] M. L. Lewis, Solvable groups whose degree graphs have two connected components, J. Group Theory, 4 no. 3 (2001) 255–275. [6] O. Manz and T. R. Wolf, Representations of solvable groups., Cambridge University Press, Cambridge, 1993. [7] P. P. P´alfy, On the character degree graph of solvable groups. I. three primes, Period. Math. Hungar., 36 no. 1 (1998) 61–65. [8] P. P. P´alfy, On the character degree graph of solvable groups. II. disconnected graphs, Studia Sci. Math. Hungar., 38 (2001) 339–355.
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On the number of disconnected character degree graphs satisfying Palfy’s inequality