On the nilpotent graph of a finite group

Torres, Jaime; Gutierrez Garcia, Ismael; Garcia Claro, Elias Javier

doi:10.22108/ijgt.2025.145208.1961

تعداد نشریات	43
تعداد شماره‌ها	1,744
تعداد مقالات	14,253
تعداد مشاهده مقاله	35,313,509
تعداد دریافت فایل اصل مقاله	14,018,615

	On the nilpotent graph of a finite group
International Journal of Group Theory
مقالات آماده انتشار، اصلاح شده برای چاپ، انتشار آنلاین از تاریخ 02 تیر 1404 اصل مقاله (476.92 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22108/ijgt.2025.145208.1961
نویسندگان
Jaime Torres¹؛ Ismael Gutierrez Garcia^* ¹؛ Elias Javier Garcia Claro²
¹Departamento de Matemáticas y Estadı́stica, Universidad del Norte, Barranquilla, Colombia
²Departamento de Matemáticas, Universidad Autónoma Metropolitana, Unidad Iztapalapa, Ciudad de México - México
چکیده
The nilpotent graph of a finite group $G$, denoted by $\Gamma_N(G)$, is a simple graph whose vertex set is $G - nil(G)$, where $nil(G)= \{g\in G: \langle g, h\rangle \ \text{is nilpotent for all} \ h\in G\}$, and two distinct vertices are related if they generate a nilpotent subgroup of $G$. In this work, lower bounds for the clique number and the number of connected components of $\Gamma_N(G)$ are presented in terms of the size of its Fitting subgroup and the number of its strongly self-centralizing subgroups of $G$, respectively. We prove that no finite non-nilpotent group has a self-complementary nilpotent graph. Furthermore, for the dihedral group $D_{n}$, it is determined that the number of connected components of its nilpotent graph is one more than $n$ when $n$ is odd or one more than the $2'$ part of $n$ when $n$ is even. In addition, a formula for the number of connected components of $\Gamma_N(psl(2,q))$, where $q$ is a prime power, is provided.
کلیدواژه‌ها
Graphs associated with groups؛ nilpotent group؛ hypercenter of a group؛ nilpotentizer؛ strongly self-centralizing subgroups

مراجع
[1] A. Abdollahi, S. Akbari and H. R. Maimani, Non-commuting graph of a group, J. Algebra, 298 no. 2 (2006) 468–492. [2] A. Abdollahi and M. Zarrin, Non-nilpotent graph of a group, Comm. Algebra, 38 (2009) 4390–4403. [3] P. Bhowal, D. Nongsiang and R. K. Nath, Solvable graphs of finite groups, Hacet. J. Math. Stat., 49 no. 6 (2020) 1955–1964. [4] T. C. Burness, A. Lucchini and D. Nemmi, On the soluble graph of a finite group, J. Combin. Theory Ser. A, 194 (2023). [5] A. K. Das and D. Nongsiang, On the genus of the nilpotent graphs of finite groups, Comm. Algebra, 43 no. 12 (2015) 5282–5290. [6] A. K. Das and D. Nongsiang, On the genus of the commuting graphs of finite non-abelian groups, Int. Electron. J. Algebra, 19 no. 19 (2016) 91–109. [7] T. S. Developers, W. Stein, D. Joyner, D. Kohel, J. Cremona and B. Er¨ocal, (2020), SageMath, version 9.0. Retrieved from http://www.sagemath.org.. [8] M. Ghaffarzadeh, M. Ghasemi, M. L. Lewis and Hung P. Tong-Viet, Nonsolvable groups with no prime dividing four character degrees, Algebr. Represent. Theor., 20 no. 3 (2017) 547–567. [9] I. Grossman and W. Magnus, Groups and their graphs, Mathematical Association of America, 1992. [10] N. Hartsfield and G. Ringel, Pearls in graph theory: a comprehensive introduction, Academic Press, Cambridge, 1990. [11] B. Huppert, Endliche gruppen I (grundlehren der mathematischen wissenschaften, 134, Springer-Verlag, Berlin, Heidelberg and New York, 1979. [12] A. Iranmanesh and A. Jafarzadeh, On the commuting graph associated with the symmetric and alternating groups, J. Algebra Appl., 7 no. 1 (2008) 129–146. [13] A. Lucchini and D. Nemmi, The non-F graph of a finite group, Math. Nachr., 294 no. 10 (2021) 1912–1921. [14] D. Nongsiang and P. K. Saikia, On the non-nilpotent graphs of a group, Int. Electron. J. Algebra, 22 (2017) 78–96. [15] Derek J. S. Robinson, A course in the theory of groups, Second Edition, Graduate Texts Math., 80, Springer-Verlag, New York, 1996.
آمار تعداد مشاهده مقاله: 81 تعداد دریافت فایل اصل مقاله: 31

سامانه مدیریت نشریات علمی. طراحی و پیاده سازی از سیناوب

پیوندهای مفید

اخبار و اعلانات

آمار

On the nilpotent graph of a finite group