A graph associated to a fixed automorphism of a finite group

A graph associated to a fixed automorphism of a finite group

Let G be a finite group and Aut(G) be the group of automorphisms of G. We associateagraphtoagroup G andfixedautomorphism α of G denotedby Γα Gasfollows. Thevertexsetof Γα G is G\Zα(G) andtwovertices x,g ∈ G\Zα(G)are adjacent if [g,x]α ̸= 1 or [x,g]α ̸= 1, where [g,x]α = g−1x−1gxα andZ α(G) = {x ∈ G|[g,x]α = 1 forall g ∈ G}. Inthispaper,westatesomebasic properties of the graph, like connectivity, diameter, girth and Hamiltonian. Moreover, planarity and 1-planarity are also investigated here.

Let G be a finite group and Aut(G) be the group of automorphisms of G. We associateagraphtoagroup G andfixedautomorphism α of G denotedby Γα Gasfollows. Thevertexsetof Γα G is G\Zα(G) andtwovertices x,g ∈ G\Zα(G)are adjacent if [g,x]α ̸= 1 or [x,g]α ̸= 1, where [g,x]α = g−1x−1gxα andZ α(G) = {x ∈ G|[g,x]α = 1 forall g ∈ G}. Inthispaper,westatesomebasic properties of the graph, like connectivity, diameter, girth and Hamiltonian. Moreover, planarity and 1-planarity are also investigated here.