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

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