For a given nonabelian finite group G and S ⊆ G\\\\\\\\\\\\\\\\Z(G), where Z(G) denotes the center of G, we introduce a new graph Γ(G, S) associated to the group G as follows: Take G\\\\\\\\\\\\\\\\(S ∪ Z(G)) as its vertex set and two distinct vertices x and y being adjacent if and only if there exists an element s ∈ S such that [x, s] = 1 = [y, s]. This paper is devoted to investigate the properties of graphs Γ(G, S) and establish some graph theoretical properties. Moreover, we describe the planarity of these graphs when |S| = 1. Also, we provide some examples of finite nonabelian groups G with the property that if Γ(G, S)=Γ(H, S) and |S| = 1 for some group H and S\\\\\\\\\\\\\\\' ⊆ H\\\\\\\\\\\\\\\\Z(H), then |G| = |H|.

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