A graph associated to a fixed automorphism of a finite group

Abstract

Let $G$ be a finite group and $Aut(G)$ be the group of automorphisms of $G$. We associate a graph to a group $G$ and fixed automorphism $\alpha$ of $G$ denoted by $\Gamma_G^\alpha$. The vertex set of $\Gamma_G^\alpha$ is $G\backslash Z^\alpha(G)$ and two vertices $x,g\in G\backslash Z^\alpha(G)$ are adjacent if $[g,x]_\alpha\neq 1$ or $[x,g]_\alpha\neq 1$, where $[g,x]_\alpha=g^{-1}x^{-1}gx^\alpha$ and $Z^\alpha(G)=\{ x\in G\,|\, [g,x]_\alpha=1\,\,\textrm{for all}\,\, g\inG \}$. In this paper, we state some basic properties of the graph, like connectivity, diameter, girth and Hamiltonian. Moreover, planarity and 1-planarity are also investigated here.