A bipartite graph associated to a finite group

Abstract

Let [Formula: see text] be a finite group and [Formula: see text] be the automorphism group of [Formula: see text] (i.e. [Formula: see text]). We associated a bipartite graph, denoted by [Formula: see text], to [Formula: see text] and its automorphism group [Formula: see text] as follows: two parts of the vertex set are [Formula: see text] and [Formula: see text], where [Formula: see text] is the set of elements [Formula: see text] such that [Formula: see text] for all [Formula: see text] and [Formula: see text] is the set of automorphisms [Formula: see text] such that [Formula: see text] for all [Formula: see text]. Two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we investigate some fundamental properties of [Formula: see text] such as connectivity, diameter, girth, Hamiltonian, independence and dominating numbers. Moreover, planarity and outer planarity of the graph are studied.