On the non-nilpotent graphs of a group

Authors : Deiborlang NONGSİANG, Promode Kumar SAİKİA

Pages : 78-96

Doi:10.24330/ieja.325927

View : 14 | Download : 15

Publication Date : 2017-07-11

Article Type : Research Paper

Abstract : Let $G$ be a group and $nilinsert ignore into journalissuearticles values(G);=\{x \in G \mid \langle x,y \rangle \text{ is nilpotent for all }\\ y \in G\}$.  Associate a graph $\mathfrak{R}_G$ insert ignore into journalissuearticles values(called the non-nilpotent graph of $G$); with $G$ as follows: Take $G \setminus nilinsert ignore into journalissuearticles values(G);$ as the vertex set and two vertices are adjacent if they generate a non-nilpotent subgroup. In this paper we study the graph theoretical properties of $\mathfrak{R}_G$. We conjecture that the domination number of the non-nilpotent graph of every finite non-abelian simple group is 2. We also conjecture that if $G$ and $H$ are two non-nilpotent finite groups such that $\mathfrak{R}_G\cong \mathfrak{R}_H$, then $|G| = |H|$. Among other results, we show that the non-nilpotent graph of $D_{10}$ is double-toroidal.  
Keywords : Non nilpotent graph, finite group