On the nilpotent graph of a finite group

Articles in Press

Document Type : Research Paper

Authors

1 Departamento de Matemáticas y Estadı́stica, Universidad del Norte, Barranquilla, Colombia

2 Departamento de Matemáticas, Universidad Autónoma Metropolitana, Unidad Iztapalapa, Ciudad de México - México

10.22108/ijgt.2025.145208.1961

Abstract

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.

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References

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International Journal of Group Theory

Articles in Press, Corrected Proof
Available Online from 23 June 2025

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History

  • Receive Date: 09 May 2025
  • Revise Date: 07 June 2025
  • Accept Date: 13 June 2025
  • Published Online: 23 June 2025

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