A bipartite graph associated to a finite group.

Let G be a finite group and A G be the automorphism group of G (i.e. Aut (G)). We associated a bipartite graph, denoted by Γ G , to G and its automorphism group Aut (G) as follows: two parts of the vertex set are G \ L (G , Z (G)) and A G \ Aut c (G) , where L (G , Z (G)) is the set of elements g ∈ G such that g − 1 α (g) ∈ Z (G) for all α ∈ A G and Aut c (G) is the set of automorphisms β ∈ A G such that g − 1 β (g) ∈ Z (G) for all g ∈ G. Two vertices g ∈ G \ L (G , Z (G)) and α ∈ A G \ Aut c (G) are adjacent if and only if g − 1 α (g) ∉ Z (G). In this paper, we investigate some fundamental properties of Γ G such as connectivity, diameter, girth, Hamiltonian, independence and dominating numbers. Moreover, planarity and outer planarity of the graph are studied. [ABSTRACT FROM AUTHOR]